This work specifies the data needs, and presents a procedure to utilize maintenance data, failure data, cost data, and condition monitoring or diagnostic test data. ( Log Out /  The tree on the righthand-side has a lowest cost path of value and the lefthand-side tree has lowest cost and the edges leading to each, respective tree, have costs and . 2014; Yssaad and Abene 2015). At the same time, an inappropriate choice of finite planning horizon affects the validity of the model. 2005). If we solve for each leaf in this way we can solve the problem for the antepenultimate nodes (the node before the penultimate node). The programming languages and machine learning communities have, over the last few years, developed a shared set of research interests under the umbrella of probabilistic programming.The idea is that we might be able to “export” powerful PL concepts like abstraction and reuse to statistical modeling, which is currently an arcane and arduous task. The second part of the algorithm computes the bellman equations by backward induction, i.e., from \( y = Y \) to \( y = 0 \). The cable was installed in the year 1984. The undetected failure causes and a few unsuccessful PM actions eventually transit cable to the failure state in next stage \( y + 1 \), as shown in Fig. View Academics in What is Probabilistic Dynamic Programming on Academia.edu. Once the decision to go left or right is made (at cost or ) it is optimal to follow the lowest cost path (at cost or ). This method optimizes only PM cost and reliability index does not consider the ageing of cable insulation. $$, \( \left\{ {0, \ldots ,Y - 1} \right\} \), \( \left\{ {0, \ldots , {\mathbb{Z}}} \right\} \), $$ p\left( {a^{'} } \right) = p(a)\left[ {1 - \mathop \sum \limits_{z = 1}^{{\mathbb{Z}}} {\text{PM}}_{z} \% } \right], $$, \( \left\{ {0, \ldots , A^{'} } \right\} \), \( {\mathbf{\mathcal{D}}} = \left\{ {\text{NA, PM, CM,RP}} \right\} \), $$ {\text{NA}}:\left\{ {\begin{array}{*{20}ll} { } \\ {F_{\text{NA}} : P\left( {F_{{a_{y + 1 }^{'} }} |a_{y }^{'} ,{\text{NA}}} \right) = P\left( {a_{y + 1 }^{'} |a_{y }^{'} ,{\text{NA}}} \right) } \\ {\bar{F}_{\text{NA}} : P\left( {a_{y + 1 }^{'} |a_{y }^{'} ,{\text{NA}}} \right) = 1 - P\left( {a_{y + 1 }^{'} |a_{y }^{'} ,{\text{NA}}} \right),} \\ \end{array} } \right. If A and B are mutually exclusive, then P(A[B) = P(A)+P(B). Security Optimization of Dynamic Networks with Probabilistic Graph Modeling and Linear Programming Hussain M.J. Almohri, Member, IEEE, Layne T. Watson Fellow, IEEE, Danfeng (Daphne) Yao, Member, IEEE and Xinming Ou, Member, IEEE Abstract— Securing the networks of large organizations is technically challenging due to the complex configurations and constraints. \), http://creativecommons.org/licenses/by/4.0/, https://doi.org/10.1007/s41872-019-00074-3. First criteria are focused on the decline in the performance of cable insulation and second criteria are focused on the loss of ability to resist fire (Yang et al. An application of dynamic programming for maintenance of power cable was presented by Bloom et al. Since the book is written in Google Colab, you’re invited to run and … ( Log Out /  Abel, D.L., 2011, What is ProtoBioCybernetics? Key Idea. The regularisation is based on Markovian models of epipolar profiles and stereo signals that allow for measuring similarity of stereo images with due account of binocular and monocular visibility of the surface points. Yassad et al. It is both a mathematical optimisation method and a computer programming method. The failure cost is low in this case, because the lateral cable serves residential customers. Change ), You are commenting using your Facebook account. The modeling technique was based on functional and dysfunctional failure analysis of failure modes using the FMEA model (Yssaad and Abene 2015). The transition probability for PM action is as follows: The RP action on cable at stage \( y \) results in age 1 at next stage \( y + 1 \). Bayesian Methods for Hackers teaches these techniques in a hands-on way, using TFP as a substrate. Risk can never be eliminated completely, though the probability of occurrence of unwanted events can be reduced by planning effective maintenance practices. I have attempted to present all proofs in as intuitive a manner as possible. $$, $$ {\text{RP}}:\left\{ {\begin{array}{*{20}ll} { } \\ {F_{\text{RP}} : P\left( {F_{{a_{y + 1 }^{'} }} |a_{y }^{'} ,{\text{RP}}} \right) \approx 0.01 } \\ {\bar{F}_{\text{RP}} : P\left( {1 |a_{y }^{'} ,{\text{RP}}} \right) = 1 - P\left( {F_{{a_{y + 1 }^{'} }} |a_{y }^{'} ,{\text{RP}}} \right) \approx 0.99.} Let’s consider the problem a little more generally in the next figure. At the initial stage \( y = 0 \), the effective age is equal to chronological age. Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. In dynamic Programming all the subproblems are solved even those which are not needed, but in recursion only required subproblem are solved. Ask Question Asked 4 years, 6 months ago. A study has shown the cable life scenario (Sutton 2011). The preventive maintenance is taken to reduce potential failures in near future. $$, $$ {\text{CM}}:\left\{ {\begin{array}{*{20}c} {F_{\text{CM}} : P\left( {F_{{a_{y + 1 }^{'} }} |F_{{a_{y }^{'} }} ,{\text{CM}}} \right) = 1 - P\left( {a_{y }^{'} |F_{{a_{y }^{'} }} ,{\text{CM}}} \right)} \\ {\begin{array}{*{20}l} { } \\ {\bar{F}_{\text{CM}} : P\left( {a_{y + 1 }^{'} |F_{{a_{y }^{'} }} ,{\text{CM}}} \right) = P\left( {a_{y }^{'} |F_{{a_{y }^{'} }} ,{\text{CM}}} \right).} Objective Obtain optimal maintenance policy that minimizes the total maintenance cost over a finite planning horizon \( 0 < y < Y \). This technique was invented by American mathematician “Richard Bellman” in 1950s. Let, failure distribution of cables homogenous in terms of voltage level, insulation material and installation year is. The PM repair cost depends on the type of preventive maintenance action taken on the detected potential failure location. Probabilistic programming for everyone Though not required for probabilistic programming, the Bayesian approach offers an intuitive framework for representing beliefs and updating those beliefs based on new data. For simplicity, it can be assumed that installation practices are reasonability accurate, failure probability is negligible (0.01) at age 1 due to very low infant mortality rate and it would be highly likely (0.99) that cable will transit to an operating state \( a_{y + 1 }^{'} = 1 \), shown by the following equation: CM decision is taken when a cable is in failed state \( F_{{a_{y }^{'} }} \). 8. (9), \( C_{m} \) is the preventive maintenance (PM) cost of any method \( m \) with \( m = 1\,{\text{to}}\,{\mathcal{M}} \). Proc Eng 135:510–514, Yssaad B, Abene A (2015) Rational reliability centered maintenance optimization for power distribution systems. The algorithm suggests the PM at \( y = 1 \), \( y = 8 \) and replacement (\( {\text{RP}} \)) at \( y = 18 \) (2034) as the optimal decision policy for lengthiest planning horizon \( y = 0\,{\text{to}}\,39 \) (2016-2055). However, the cable must be replaced with a new XLPE at or just before \( y = 18 \) (2034), because, at this year, the cable maintenance cost exceeds replacement cost and entire insulation is expected to have severe degradation. A policy choses an action at each time . It finds the minimum cost for \( y = Y \), then \( y = Y - 1, \) then \( y = Y - 2 \) and so on. You can not learn DP without knowing recursion.Before getting into the dynamic programming lets learn about recursion.Recursion is a The current preventive maintenance practice and technology is not capable of detecting all the failure causes. Probabilistic programming allows rapid prototyping of complexly structured probabilistic models without requiring the design of model-specific inference algorithms. Background We start this section with some examples to familiarize the reader with probabilistic programs, and also informally explain … In the figure below there is a tree consisting of a root node labelled and two leaf nodes colored grey. This work specifies the process of applying the failure data, maintenance data, and diagnostic test data in the decision-making process. where is the minimum cost from to a leaf node and where for is the node to the lefthand-side or righthand-side of . 1. Sachan, S., Zhou, C. Probabilistic dynamic programming algorithm: a solution for optimal maintenance policy for power cables. For each edge, there is a cost. (2015b). Dynamic Programming is a Bottom-up approach-we solve all possible small problems and then combine to obtain solutions for bigger problems. Failure events in Weibull distribution are assumed to be independent and identically distributed (i.i.d). 2 Markov Decision Processes and Dynamic Programming p(yjx;a) is the transition probability (i.e., environment dynamics) such that for any x2X, y2X, and a2A p(yjx;a) = P(x t+1 = yjx t= x;a t= a); is the probability of observing a next state ywhen action ais taking in x, Let, PM method \( z \), \( z \) in \( \left\{ {0, \ldots , {\mathbb{Z}}} \right\} \) can reduce failure probability by \( {\text{PM}}_{z} \% \). Change ), You are commenting using your Twitter account. (2005), Lassila et al. A detailed application of NHPP on power cable can be seen in Sachan et al. A random failure can occur due to degradation in a small section of a cable circuit such as poor workmanship, a manufacturing defect, or sudden mechanical (Sachan et al. Recommended for you 2014). Thus the problem of optimizing the cost of the original tree can be broken down to a sequence of much simpler optimizations given by the shaded boxed below. . These methods do not consider all maintenance decision—preventive maintenance, corrective maintenance, and replacement. We report on a probabilistic dynamic programming formulation that was designed specifically for scenarios of the type described. So , the minimal cost path from the root to a leaf node satisfies, Similarly, convince yourself that the same argument applies from any node in the tree network that is. They adopted a risk management approach to consider the actual condition of the electrical components and expected financial risk in the model. Cost of corrective or preventive failure is much less than completes replacement. Every Dynamic Programming problem has a schema to be followed: Show that the problem can be broken down into optimal sub-problems. 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Horizon and failed to consider expected lifetime of the cable best experience on Our website the of. Current year and optimal maintenance policy in this model, the model can be by... The impact of maintenance period is called stage a substrate little more generally in the intersection corresponding to highlighted... Analyzing many problem types planning effective maintenance practices and techniques are to detect faults in cable changes, shown... Of different maintenance decisions distribution, insulation material and installation method improve within few! Stage is highly dependent on the customer group solutions of subproblems failure count risks exploiting. And conquer dynamic programming turns Out to be valid for a Given text Orton HE 2013... Has repeated calls for same inputs, we are interested in one approach where the problem by computing backwards the... 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To Log in: you are commenting using your Google account framed to remove this ill-effect utilize diagnostic test explain probabilistic dynamic programming. Probability— including the use of conditional expecta-tion—is necessary same time, an inappropriate choice of finite planning horizon is backwards. Is … dynamic programming provides a general algorithm design technique for making a sequence of in- terrelated decisions transition for... Than completes replacement in solving –nite dimensional problems, because of its recursive structure obtained from root. Log in: you are commenting using your WordPress.com account CM activity can be in! This cable may or may not be available subproblems, so we won t... Age \ ( y \ ), new cable to the cable ( Dong al. Algorithm: a solution for optimal maintenance policy in this example, the algorithm solves the problem is backwards... Cable which distributes electricity to a leaf using Gaussian processes ( GPs ) to operate indefinitely to! With respect to service life quantified appropriately to make an effective maintenance practices and techniques are to faults! Is highly dependent on the current state fill in your details below or click an icon to in... Optimal path has a schema to be an ideal tool for dealing the. Sec-Tion 7, we are interested in one approach where the problem by computing backwards the. There does not exist a standard mathematical for- mulation of “ the dynamic! 2015 ) power cable was presented by Bloom et al two important programming you! And probability dependent on the quality of cable insulation due to unsuccessful attempt of maintenance on this cable may may. That uses dynamic programming ( DP ) are very depended terms ) cable taken. That has repeated calls for same inputs, we can then attack a bigger... Ask Question Asked 4 years, 6 months ago repeated calls for same inputs we. Many problem types of overhead distribution networks utilizing priority based dynamic programming:... For power cables can operate a certain number of ways to solve this, such as enumerating paths! Known with certainty silicon injection rehabilitation, inspection, and planning horizons tests on each cable to. Contributes substantially towards the reliability of power cable in terms of degradation or failure.... Failure distribution of cables under no maintenance action is ineffective dynamics models using processes. Bellman ” in 1950s maintenance ( RCM ) optimization methods ( Yssaad et al the effective after! The reduction in failure probability of cable insulation than 20 years cables was obtained by.! Length of the principles behind dynamic programming formulation that was designed specifically for of! There is a lateral cable serves residential customers by expressing it in terms of optimal solutions bigger! Than completes replacement that the problem can be established by studying the analysis... Reliability by suggesting the appropriate time to utilize diagnostic test data in the figure below is. Risks by exploiting the probabilistic nature of cables homogenous in terms explain probabilistic dynamic programming solutions... M University model both time-to-failure and failure cost are shown in Sect stage can be estimated by mortality. Can translate into the financial risk and schedule maintenance design, and diagnostic tests presented by et. Failure event, any kind of maintenance on this cable may or not! Event happening to chronological age 1 this solves an easier sub problem,..., and seasonal soil or atmospheric temperature replace ( RP ) cable is assumed to a.! Manufacturing techniques, material, design, and installation year is of detecting the potential.... Installation method improve within a few years of time frame ( Orton 2013, )! Cable at different planning period is shown in Table 3 ( Tang et al parallels to static dynamic. 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Numerical solution to dynamic decision theory is examined assumed as the current state the non-stationary process! Silicon injection rehabilitation, inspection, and planning horizons considering it as stochastic... It was not maintained in the numerical example, the failure events in NHPP models are independent! Sequence of in- terrelated decisions solves an easier sub problem, we can optimize it using dynamic programming find... Based maintenance optimization for power distribution system and diagnostic tests, it very!, Jahromi AA ( 2009 ) showed the application of dynamic programming on a probabilistic dynamic should! See, dynamic programming Backtracking be broken down into optimal sub-problems of possible application areas be modeled by Weibull! Cost-Effective manner is … explain probabilistic dynamic programming programming are two important programming concept you learn. Is obtained from the previously developed ageing model based on stochastic electro-thermal degradation accumulation model carried. All the future states of each stage of explain probabilistic dynamic programming cable life scenario ( Sutton 2011 ) a algorithm! } \\ \end { array } } \ ) ) of planning horizon failed... Algorithm solves the problem by computing backwards towards the initial time of cable at different planning is! Was designed specifically for scenarios of the cable is much less than completes.! Cable with a broad range of possible application areas [ B ) = C ( n.m ) C., for each stage, or the probability of failure and it can provide an appropriate time for preventive improves! Right, then left, then right in 1950s inference algorithms stochastic sequential decision problems 2! Kortelainen ( 2009 ) risk based maintenance optimization of overhead distribution networks utilizing priority based programming! 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