Dynamic Bayesian Networks model to estimate process availability

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Dynamic Bayesian Networks model to estimate process availability

Philippe Weber

To cite this version:

Philippe Weber. Dynamic Bayesian Networks model to estimate process availability. Sep 2002, pp.184- 189. �hal-00128463�

Dynamic Bayesian Networks model to estimate process availability.

Weber P.

Centre de Recherche en Automatique de Nancy (CRAN), CNRS UMR 7039 Université Henri Poincaré, Nancy 1

2, rue Jean Lamour

F-54519 VANDOEUVRE-LES-NANCY Cedex – FRANCE Email: [philippe.weber@esstin.uhp-nancy.fr]

Abstract

The prospective work reported here explores a new methodology to develop Dynamic Bayesian Network-based Availability and prognosis for manufacturing processes. This work is justified with the complex systems by the need of controlling and maintaining in dynamical way the global system performances in order to optimise the maintenance strategies. The added value of our methodology is to formalise the dynamic models by means of Dynamic Bayesian Networks (DBNs). The feasibility of this methodology is tested in a simple case of the system availability estimation comparing DBN model with the classical Markov chain model.

Keywords: Dynamic Bayesian Network, Markovian process, Markov chain, Availability.

1. INTRODUCTION

One of the main challenges of the Extended Enterprise is to maintain and optimise, in dynamics, the quality of the services delivered by industrial objects all along their life cycle. In that way, to control the direct and indirect costs of the system compared with the necessary availability, is a first challenge raised. The objective is thus to have maintenance processes mainly in terms of decision-making aids in order to guarantee maximum components availability keeping the system in operation.

Nowadays, most of current automated systems do not provide the means for intelligent interpretation of information copying with large process disturbances and for predicting the consequences of a future action.

However tools issued from Artificial Intelligence are used now to bring a first decision-making aid for manufacturing systems. Recent works on system safety and Bayesian Networks (BNs) are developed [6]. Moreover, Bouissou propose [2], within the SERENE project, a hierarchical decomposition of the decision-making model for system safety analysis. As for them, Bobbio [1] explain how the Fault Tree can be achieved using BNs. Using the Object Oriented

Bayesian Networks (OOBNs) concept defined by Koller [8] the authors Weber et al. propose a model based decision. The model allows the fault diagnosis or availability estimation to compare several maintenance strategy ([12] and [13]). The solution proposed in these last papers are based on a static modelling of the system.

In order to add the dynamic dimension of this model we propose to introduce Dynamic Bayesian Networks as an equivalent model to the Markov chain [4] [11]. Thus, this paper is dedicated to the comparison between Markov chain and the Dynamic Bayesian Network. The proposed methodology has originality on formalising, by means of DBNs (section 2), a model of component availability (section 3). To show the feasibility of this prospective methodology an simulation of a classical serial system is developed (section 4) to estimate the availability of the system. Section 5 presents the conclusions and prospects.

2. BAYESIAN NETWORK THEORY

The Bayesian Networks (BNs) are probabilistic networks based on graphs theory. Nodes represent the various variable of the system and the edges which

connect them indicate the dependency between variables. Variables are defined over several states.

The dynamic bayesian networks allow to take time into account defining different nodes to model the variable states in regards to different time slices.

2.1. The Bayesian Network notations

The BNs are Directed Acyclic Graph (DAG), they are used to represent uncertain knowledge in Artificial Intelligence [5]. A BN is defined as a pair: G=((N, E),P), where (N,E) represents the DAG; “N” is a set of nodes; “E” is a set of directed edges E⊂N×N, describing the dependencies between nodes. P represents the set of Conditional Probability Tables (CPT) which quantify the dependency between variables. In this work, the BN represent discrete random variables, and each nodes are defined over different states.

A discrete random variable X is represented by a node n∈N with a finite number of mutually exclusive states. The states are defined in a state space

{

n ⁿK

}

n k

n: s₁,...s ,...s

S . The vector D(n) denotes a

probability distribution over these states:

[

^{p p}_k ^p_K

]

n) ... ...

( = ₁

D

≥0

pk and 1

1

∑ =

= K

k pk (1)

Where p_k is the probability of n being in state s_kⁿ defined in the following as:

) ( ⁿ_k

k n s

p =P = (2)

The nodes n_i and n_j are linked by a directed edge, if the pair

(

ⁿi,ⁿj

)

∈^E and

(

ⁿj,ⁿi

)

∉^E. n_i is then considered as the parent of n_j. The set of parents of a node n is defined as π

( )

n . If ^π

( )

^{n =φ, then n is}

a root node (^π

( )

n_i ^=φ, and π

( )

nj =ni).

ni nj

Fig. 1. A basic BN.

In pair G, each node of N is associated to a Conditional Probability Table (CPT) of the set P. The CPT quantify the dependencies between random variables as conditional probabilities.

The nodes n_i and n_j (Fig. 1.) are defined over the states i

{

^{i i}

}

nK n : s₁n ,...s

S and j

{

^{j j}

}

n L n n : s₁ ,...s

S . The a

priori probability P

(

ni =skⁿⁱ

)

allows to define the CPT of the node n_i because n_i is a root node:

ni

s₁ P(n_i=s₁ⁿⁱ)

… …

ni

sk P(n_i=s_kⁿⁱ)

… …

ni ni

sK P(n_i=sⁿ_Kⁱ) Tab. 1

This conditional probability table is written as a vector:

( )

T

n K i

n i i

i i

s n

s n n

















=

=

=

) (

...

)

( ₁

P P

P (3)

Where T denote the transposition operator.

The CPT of n_j is defined by the conditional probabilities P

( )

njni over each n_j state knowing the state of n_i (Table 2).

nj s₁^nj s_l^nj sⁿ_L^j

ni

s₁ P

(

nj =s₁^nj n₁=s₁ⁿⁱ

) (

^j i ⁿⁱ

)

n L

j s n s

n = = ₁

P

i …

n

ni

sK

(

^j i ⁿKⁱ

)

n

j s n s

n = ₁ =

P

(

^j i ⁿKⁱ

)

n L

j s n s

n = =

P Tab. 2 CPT of the node n_j

The CPT of n_j is defined by the conditional probabilities P

( )

njni over each n_j state knowing the state of n_i (Table 2).

Knowing its parents π

( )

nj , the CPT of n_j is defined as a matrix P(n_jπ(n_j)):

( )

( ) ( )

( ) ( )

_^_











=

=

=

=

=

=

=

=

= π

j i j i

j i j i

Kn i n L n j

K i n j

i n n L n j

i n j

j j

s n s n s

n s n

s n s n s

n s n

n n

P P

P P

L

M M

L

1

1 1

1

) ( P

(4)

In a BN, a probability is allowed to each state of the nodes. These probabilities are defined, a priori, for root nodes and computed by inference for the others.

The computation is based on the probabilities of the parents states and the CPT.

The probabilities P(n_j=s₁^nj) is computed as follows:

( )

( ) ( )

(

^{j i}

) (

ⁱ

)

i j i

j

n K i n K i n j

i n i n

n j n j

s n s n s n

s n s n s n s n

=

⋅

=

= +

+

=

⋅

=

=

=

=

P P

P P

P

1

1 1

1 1

... (5)

Then for each n_j states the equation (5) leads to the following matrix relation:

( )

nj =D

( )

ni ⋅P

(

nj π⁽nj⁾

)

D (6)

And a generalisation of (6) is:

( )

nj =D

⁽

π⁽ni⁾

⁾

⋅P

(

nj π⁽nj⁾

)

D (7)

where D

(

π⁽nj⁾

)

represents the vector of the probability distribution over all combination of the states of each n_j parent’s nodes.

Moreover, the most important use of BN is in revising probabilities according to the actual observations of events. In this way, inference computes the probabilities allowed to a node state, knowing the state of one or several variables. If the knowledge about events is unavailable then the computation is based on a priori probabilities.

However if the knowledge is increasing, the computation takes it into account and the results are adapted to the process state. It is then possible to estimate the impacts of a random variable on the other variable of the process modelled as a BN.

Knowledge is formalised as evidence, (hard) evidence (instantiation) of the random variable X in a

BN is evidence that the state of the node n∈N is one of the states

{

n ⁿK

}

n k

n: s₁,...s ,...s

S . For instance X is in state s₁ⁿ:

1 ) (n=s₁ⁿ =

P and P(n=s_kⁿ_≠1)=0 (8) Nevertheless knowledge can be uncertain. Then, the notion of soft evidence is introduced. The soft evidence for a node n is any evidence that enables to update the prior probability values for the states of n. The soft evidence is an instantiation of the node representing the random variable using knowledge as a distribution over the set

{

n ⁿK

}

n k

n: s₁,...s ,...s

S . For

instance X is in state s₁ⁿ and sⁿ_K with the same probability and not in the other states:

5 . 0 ) (n=s₁ⁿ =

P , P(n=sⁿ_K)=0.5

and P(n=s_kⁿ_{≠( )1,K} )=0. (9)

2.2. Dynamic time-sliced Bayesian Network A Dynamic Bayesian Network (DBN) is a extended version of the Bayesian Network, including temporal information. DBN provides a representation of the different states of a process based on a set of time-indexed random variables. Then random variable Xi is represented at the time t by a node n_(i,t)∈N with a finite number of states _i

{

ⁱ nⁱ Kⁿⁱ

}

k n : s₁n,...s ,...s

S .

( )

ni_,t

D denotes a probability distribution over these states at the time t. Several stages of the time are represented by several sets of nodes ^N

( )

^t₁ ,…^N

( )

^t_τ .

( )

^t₁

N defined the random variables in a time slice t₁. Directed arcs between time slices represent the temporal dependence between a variable and its states at different future time slices. Then DBN models random variables and their impacts over the future distribution of other variables.

Defining these impacts as transition-probabilities between the states of the variable at (t) and at (t+1), these transition-probabilities lead to defined an inter- time slices conditional probability tables equivalent to the CPT defined in the previous section.

( )it

n _,

(i,t+1) n

)) ( (n_i,t+1π n_i,t+1 P

Fig. 2. A DBN for the random variable X_i with 2 time slices.

The P(n_i,t+1π(n_i,t+1)) respects the Markov properties means that the future is conditionally independent of the past given the present [7].

Moreover this CPT P(n_i,t+1π(n_i,t+1)) is equivalent to a markovian model of the variable X_i (Figure 2).

Starting from a observed situation at time t, the probability distribution D(n_i,t+1) over n_i states is computed by the inference of the DBN.

To compute the probability distribution D(n_i,t+T), several solutions are proposed in the literature. One of them consist in developing T time slices [7], another solution is based on iterative inferences which is the solution developed in this paper.

The inference is key point in the DBN because it allows to introduce the notion of time. It is so possible to calculate the behaviour of the distribution of X_i over its states at the time (t+dt) knowing the states distribution at time (t). The states distribution at (t+2dt)… are computed using successive inferences.

To perform the successive inferences, input nodes and output nodes are defined. The input nodes are nodes allowing to describe the observation in the time slice (t). The output nodes represent the result on the variables after inference in the time slice (t+dt). To initialise the iterative inference process, the output node results are considered as the new observations.

Then soft evidence are used to initialise n_(i,t) through a time feedback.

( )^it n _,

(i^,t+¹) n time feedback

inference

)) ( (n_i,t+1π n_i,t+1 P

Fig. 3. A DBN for the random variable X_i with 2 time slices.

3. DYNAMIC BAYESIAN NETWORK TO REPRESENT A COMPONENT AVAILABILITY

3.1. Classical Markov Chain model

To model the component availability, the classical method used is based on Markov Chain theory. This method leads to a graph (Fig. 4), considering the component as a random discrete variable X_i with two states Sn:

{ }

s₁ⁿ,s₂ⁿ . The state s₁ⁿ represent that the

component X_i is available, and the state s₂ⁿ represent that the component is unavailable.

The matrix Q represents the transition probability between the different states s₁ⁿ and s₂ⁿ.



 





µ

− µ

λ λ

= −

Q (10)

As it is presented in figure 4, different nodes are defined to model the random variable at several times.

It is on the Dynamic Bayesian Networks principle that the Markov model of the component X_i availability and unavailability is translated to a Bayesian Network model.

µ λ

s₁n s₂ⁿ

Fig. 4. A Markov Process for the random variable X_i.

3.2. Dynamic Bayesian Network model

To represent the Dynamic Bayesian Network equivalent of the Markov process, two nodes are defined:

- the node n_(i,t) represents the component at time slice t,

- the node n_(i,t+dt) represents the component at time slice (t+dt).

A directed edge links the node n_(i,t) to n_(i,t+dt). This edge represents the dependency of the component states at the time slice (t+dt) to the states of the component at the time slice (t).

Conditional probability tables are associated to each nodes. They are used to compute the probability distribution over the variable states at each inference.

For instance, between two variable with two states, it will be necessary to use a table of 4 values. Assuming that dt =1, a discrete representation of the Markov chain model make it possible to define an equivalent DBN. The CPT of n_(i,t+1) performing a Dynamic Bayesian Network equivalent to the Markov model is defined as:

Q I

P(n_i,t+1π(n_i,t+1))= − (11) The equation (11) defines the CPT linking the two time slices. Thus these parameters represent the transition rate between the states of the variable X_i.

Then the parameters are those defined to build the Markov model of the component availability.

Then Markov model and Dynamic Bayesian Network model are equivalent.

To calculate P(n=s₁ⁿ,t+1) the probability that the variable X_i is in the state s₁ⁿ at (t+1), the following equation is used:

)) , ( ), , ( ( ) 1 ,

(n=s₁ⁿ t+ = f Pn=s₁ⁿ t Pn=s₂ⁿ t

P (12)

This probability depends on the state of X_i at the time t. The result is based on the equation (5) :

( )

( ) ( )

(

^{n s t n s t}

) (

^{n s t}

)

t s n t s n t s n t s n

n n

n

n n

n n

, ,

1 ,

, ,

1 , 1 ,

2 2

1

1 1

1 1

=

⋅

= +

= +

=

⋅

= +

=

= +

=

P P

P P

P

(13)

Leading to:

( )

(

^{n s t}

) (

^{n s t}

)

t s n

n n

n

, ,

) 1 ,

2 1

1

=

⋅ µ +

=

⋅ λ

= +

=

P P

- (1

P (14)

The equation (14) corresponds to the classical formula of the discrete model of the Markov chain given in figure 4.

4. APPLICATION :

The Figure 5 depicts graphically the Markov chain model of a system composed by two components A and B in series. As it is shown by the figure, to model this system, for which two components can not fall out of order at the same time, different states s₁ⁿ, sⁿ₂ and

s₃n are used:

• The state s₁ⁿ represents that the system is available this state corresponds to the case that the component A and the component B are both available.

• The state s₂ⁿ represents that the system is unavailable due to the failure of the component A (A is unavailable and B is available).

• The state s₃ⁿ represents that the system is unavailable due to the failure of the component B (A is available and B is unavailable).

The transition matrix Q defined the probability related to the different states s₁ⁿ, sⁿ₂ and s₃ⁿ.

















µ

− µ

µ

− µ

λ λ λ + λ

−

=

2 2

1 1

2 1 2 1

0 0 )

(

Q (15)

µ1

λ1

s₁n

s₂n

µ2

λ2 s₃ⁿ

Fig. 5. Markov chain modelling a serial system.

An equivalent model of this Markov chain realised by means of a Dynamic Bayesian Network is presented figure 6. The node AB1 represents the system at the time t and regroups the states of variables A and B. This variable contains three different states which correspond in three possible configurations which are: system is available (s₁ⁿ), system is unavailable due to A (s₂ⁿ), and system is unavailable due to B (s₃ⁿ). System can then evolve according to the probability of every state which correspond to the values of the failure rates and repair rate of the two components. The result of this degradation is modelled by the variable AB2

representing the state of the system at the time t+1.

The probability of the states of the system ‘sys’ and the components A2 and B2 can be extracted without any difficulty by a simple shaping of values. These variable have two states (available or unavailable).

Bayesian Network Structure

AB₁

AB₂

sys A₂ B₂

A(t+1) Available Unavailable

System States time t

System States time t+1

B(t+1) Available Unavailable System (t+1)

Available Unavailable

Fig. 6. Equivalent Bayesian network to the Markov chain of the serial system.

AB2

AB1 System

Available A

Unavailable B

Unavailable System

Available 1-(λ₁+λ₂) λ₁ λ₂ A

Unavailable µ1 1-µ1 0

B

Unavailable µ₂ 0 1−µ₂

Tab. 3 CPT of AB2

The CPT used to estimate the dynamic behaviour of the system availability is defined in Table 3 (from eq. (11) and (15)). This table corresponds to the CPT of the node AB2. The Figure 7 and 8 represents a system, where components A and B are identical. The parameters used are : λ₁=λ₂=0.01 and µ₁=µ₂=0.03.

Results obtained by means of DBN are identical to those obtained with the Markov Chain model.

The inferences are realised by means of program using an iterative procedure. The Matlab Bayes Net Toolbox (BNT) [10] developed by Murphy [9] is used to compute the inferences of the DBN in order to simulate the availability behaviour of the component.

The software Supercab+ [3] is used to simulate the Markov Chain model. The system availability is estimated and the asymptotic availability is equal to 0.6 after 200 inferences. For the components the asymptotic availability is equal to 0,8.

0 20 40 60 80 100 120 140 160 180 200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

System Availability

inferences

Available Unavailable

Fig. 7. DBN estimation of the system availability.

0 20 40 60 80 100 120 140 160 180 200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Availability of A or B

inferences

Available Unavailable

Fig. 8. DBN estimation of the Components A and B availability.

5. CONCLUSION AND FURTHER WORK

The proposed method is based on the Dynamic Bayesian Networks theory and carried out that it is possible to build easily the structure of a DBN model representing the availability of a system. We showed the correspondence that it is possible to establish among Markov chain and DBN.

Our working perspectives are now to solve problem of complex systems with many components.

In this case the Markov Chain are constituted by an important set of nodes and the transition matrix has a great dimension. Our objective is to decompose the model in independent sub-systems, so that the problem is more simple.

BNs constitute a powerful tool for decision- making aid in maintenance. The taking into account of the economic aspect is also possible in the BN formalism while adding to the network, nodes of costs (e.g. costs related to the failure) and nodes of decision (e.g. strategy of maintenance).

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//http.cs.berkley.edu/~murphyk/Bayes/bnt.html, (2002).

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