dynamic bayesian network for decision aided ...

DYNAMIC BAYESIAN NETWORK FOR DECISION AIDED DISASSEMBLY PLANNING Luminita DUTA1, Sidali ADDOUCHE2 University Valahia of Targoviste, 2 Carol Avenue, ROMANIA [email protected] IUT Montreuil, Université de Paris 8, 140 Rue de Nouvelle France, FRANCE [email protected]

Abstract. Disassembly process of used manufactured products is subjected to uncertainties. The optimal disassembly level that minimizes the costs of this process and maximizes the end of life components values is hard to establish. In this work, we propose a method to find influences and causalities between the main disassembly performance indicators in order to decide the optimal disassembly policy. The proposed model highlights the temporal dependencies between variables of the system and is validated using the Bayesia Lab software. In the final part of the article, the results of method implementation on a reference case study are presented so as to demonstrate the performance of our approach. Keywords. decision aid, disassembly planning, dynamic bayesian networks

1

Introduction

In recent years, the interest for profitability of the disassembly process and in managing the end of life products has considerably grown. Remanufacturing, reusing and recycling are options for re-integrating the parts of the used product in manufacturing chain of a new one. Disassembly is the central stage of the three-R processes referred as “product reconstruction” by Pearce [1]. Through this process components and materials are obtained in view of their valorization. Complex components are frequently considered as items to be disassembled. Disassembly can be a nondestructive process (addressing to parts rather than materials) or a destructive one (addressing on materials rather than items). After nondestructive disassembly, the items are sold, reused, recycled, stored for future use, or disposed of. Similarly, after destructive disassembly (dismantling), materials are either recycled or disposed of. The objective of disassembly planning is to identify the sequence of disassembly operations that will maximize the expected returns from the processed items. This is a decision making process. To maximize the end of life components revenue, an optimal disassembly policy must be applied. In this context, identifying the optimum sequence for disassembly operations is one of the most important objectives of disas-

sembly planning. The optimal disassembly policy is the one that minimizes the process costs and maximizes the final profit. However, a problem occurs: in the disassembly process of the used products, one could not know the state of the product component before the disassembly proceeds. Moreover, if the end-of-life revenues are deterministic and known, the operative costs as well as the operative disassembly times are stochastic and unknown at the beginning of the process. These values vary between certain values but have a probabilistic distribution. In this paper, an approach that integrates uncertainty in the model of components estate and stochastic values for disassembly operative costs and times is proposed. This approach takes into consideration the influences and the causalities between the main disassembly performance indicators in order to minimize the costs of the process and to maximize the end of life components values.

2

State of the art

For disassembly planning and disassembly policy optimization a large variety of methods has been used. The literature is very rich from this point of view and in the book of Lambert and Gupta [2] a complete state of art is made in respect of the disassembly sequencing and planning optimization. In this paper we will review only the works that dealt with the probabilistic disassembly planning or disassembly policies under uncertainties. One of the first articles on the probabilistic disassembly planning is [3]. The authors used the probabilistic inference mechanism of a Bayesian network. They obtained a static and a dynamic mathematical model based on the product topology, deterioration processes, and alternative disassembly methods each of which is represented by means of a random variable in a Bayesian Network. However, the mathematical model proves to be very complicated for a complex product with more than 10 components. Dependences between bayesian probabilities are hard to calculate. Grochowski [4] used hybrid bayesian networks to estimate the optimal disassembly policy. He describes an expert system, consisting of a Disassembly Petri network and a Hybrid Bayesian network, for optimal disassembly planning. The expert system identifies a component defect and proposes a disassembly policy. It doesn’t take into account the operative times or costs. Bayindir Z.P. and al [5] investigated the desired level of the recovery under various inventory control policies when the success of recovery is probabilistic. All used and returned items go into a recovery process that is modeled as a single stage operation. The recovery effort is represented by the expected time spent for it. The effect of increasing recovery effort on the success probability together with the unit cost of the operation is included by assuming general forms of dependencies. Graphical models are used to represent the process of decision making in a process which variables are probabilistic. For example, adding specific nodes to a BN such as nodes of utility and nodes of decision, one can take into account the associated costs

and uncertainties.. This modeling technique, described by Jensen in [6], uses the concept of Influence Diagram (ID). These are graphs that take into account several scenarios (decisions) and make possible to evaluate the impact of decisions on the process costs. The work of Clemen and al. [7] presents such influence diagrams in parallel with the corresponding decision trees. These two graphical methods resemble from the calculation point of view, but the representation using ID is more compact and more easily applicable in decision aid and large solution space problems. Alternative to recovered items, demand is satisfied by new items. In the field of dependability, recent studies based on modeling of industrial processes using the theory of (Bayesian Network) BN have been developed by Godichaud et al. [8]. His work demonstrates the relevance of the BN as a tool for modeling industrial processes functioning in the presence of uncertainties. He proposed an original use of DBN for decision aid in the dismantling processes subjected to uncertainties. This uncertainty relates mainly to the requirements in parts resulting from dismantling process and to the arrivals of end of life products. He also proposed end of life destinations and certain dismantling levels through a forecast of the recycled spare parts supply. However, he didn’t take into account any cost. Alami [9] treated the problem of uncertainty on the availability of recycled spare parts using an economical approach. His model of differential equation allows the evaluation of costs and revenues of the reconditioned spare parts obtained after the disassembly process. Unfortunately, the formalism of Alami has no long term effect. It is only a tool for static resolution that must be done periodically in time. The aim of our work is to incorporate the dynamic approach into the disassembly planning process by using the Bayesian formalism so as to obtain minimum process costs in real time and to update the task assignment at each moment taking into account the type of operation. In the same time, we are looking to combine the dynamic inventory with Bayesian Network learning so as to obtain a dynamic model close to the real system. To accomplish this aim one has to use the Dynamic Bayesian Networks, a well suited modeling instrument for complex systems submitted to uncertainties.

3

Dynamic Bayesian Networks

A Bayesian Network (BN) is an appropriate graphical method for modeling of causal processes and probability-based knowledge representation under uncertainty. A BN is a directed acyclic graph whose nodes represent random variables and links define probabilistic dependences between variables. These relationships are quantified by associating a conditional probability table with each node, given any possible configuration of values for its parents.

Bayesian networks have the ability of capturing both qualitative knowledge (through their network structure), and quantitative knowledge (through their parameters). While expert knowledge from practitioners is mostly qualitative, it can be used directly for building the structure of a Bayesian network. In addition, data mining algorithms can encode both qualitative and quantitative knowledge and encode both forms simultaneously in a Bayesian network [10]. The static Bayesian network can be extended to a Dynamic Bayesian Network (DBN) model by introducing relevant temporal dependencies that capture the dynamic behaviors of the system at different times. Two types of dependencies can be distinguished in a dynamic Bayesian network: contemporaneous dependencies and noncontemporaneous dependencies. Contemporaneous dependencies refer to arcs between nodes that represent variables within the same time period. Noncontemporaneous dependencies refer to arcs between nodes that represent variables at different times [11]. The advantage of DBN over Markov chains is that a DBN is a stochastic transition model factored over a number of random variables, over which a set of conditional dependency assumptions is defined. Time invariance ensures that the dependency model of the variables is the same at any point in time. While a DBN can in general represent semi-Markovian stochastic processes of order k-1, providing the modeling for k time slices, the term DBN is usually adopted when only two time slices are considered in order to model the system temporal evolution. That’s why such models are also called Two time Bayesian Networks (2-TBN) or 2-time-slice temporal Bayesian networks [12]. Each time-slice contains a set of (time-indexed) random variables, some of which are typically not observable. When a first order Markov process assumption holds the future slice at time t +1, it is conditionally independent of the past ones given the present slice at time t. In this case, it is sufficient to represent two consecutive time slices called the anterior and the ulterior layer to represent the network. However, to specify the entire network and to correctly model the system next parameters have to be provided: - the prior probabilities for root variables at time t=0; - the intra-slice conditional dependency model, together with the corresponding conditional probabilities; - the inter-slice conditional dependency model and the transition model, which explicit the temporal probabilistic dependencies between variables. - nodes from the anterior layer must contain only variables having influence on the same variable or on another variable at the ulterior level. - the inter-slice edges connecting a variable in the anterior layer to the same variable in the ulterior layer are temporal arcs;

4

Proposed model

4.1

Notations and assumptions

The following notations are used in text as well as in the graphical model: Time State Rate Utility OpTime(t) OpTime(t+1) Cost(t) Cost(t+1) Revenue(t)

The time variable represented by 20 unities State of the component at arrival Percentage of reusable/reutilization The result of decision in terms of profit Operative time at the first slice of time t Operative time at the second slice of time t Operative cost at the first slice of time t Operative cost at the second slice of time t End of life revenue at the first slice of time t End of life revenue at the second slice of Revenue(t+1) time t Constructing the model leads us to make the following assumptions: - the model is conceived for one type of product - two type of operations are possible: proper disassembly and destructive disassembly (disassembly and dismantling) - two types of end of life options are taken into consideration: reuse and recycling - the initial values (first slice of time t) for components end of life revenues, operative times and costs are deterministic and are variable into known internals - the final values (second slice of time t+1) for disassembly operative times and costs are stochastic/probabilistic since they depend on the state of the component which is not known in advance (at slice of time t) - probabilistic relations were provided for each node, expressing the probabilities of that node taking on each of its values, conditioned on the values of its parent nodes. - only operative costs are taken into consideration since the value of other costs is negligible - all costs and operative times follow a Normal distribution. - state of the component determines the rate of utilization and so the final revenue - do possible decisions are to be taken into consideration taking into account the state of the component : disassembly or dismantling - the decision is taken so as to maximize the final profit: the difference between end of life revenues and costs of the process A Dynamic Bayesian Network (DBN) model is designed to highlight the temporal dependencies and causalities between variables of the system (Fig. 1). The dynamic behavior of the disassembly process is well described modeling its stochastic variables: operative times, operative costs, type of operation, components end of life states.

Fig. 1. Dynamic Bayesian Network of the model

The aim of our work is to find the best disassembly decision at the slice time t+1 and the best combination between recyclable and reusable parts so as to minimize the costs and to maximize the final profit. Between two slice of time there are temporal arcs (Fig. 1) and the evolution of corresponding temporal nodes is followed in time during the simulation. The utility of the decision is calculated with a deterministic formula depending on the process cost, end of life revenue and rate of utilization at the slice time t+1. In the following paragraph the proposed model is validated using the Bayesian Lab software.

5

Validation and results

5.1

Case study

We present the results of method implementation on a reference case study from literature to successfully demonstrate the performance of our approach. The case study is referring to a disassembly of a notebook [13]. The case of three years old product is considered. Operative times are given in Table 1, the, disassembly operative costs are given in Table 2, the end-of-life values and the reusable/recycling percentage are specified in Table 3. Some data is taken from [14] and other is calculated.

Table 1. Operative disassembly/dismantling times

Components [A] LCD Monitor Type I [B] LCD Monitor Type II [C] Motherboard Type I [D] Motherboard Type II [E] Processor [F] Memory [G] Hard drive 20Gb [H] Hard drive 30Gb [I] CD Drive [J] Combo Drive [K] Network Card [L] Modem [M] Keyboard [N] Battery [O] Power Adaptor

Disassembly Time (s) 15 15 20 20 5 4 8 8 6 6 4 5 8 5 4

Dismantling Time (s) 8 8 10 10 3 2 4 4 3 3 2 3 4 3 2

Table 2. Operative costs


Disassembly Cost ($) 0.25 0.25 0.15 0.15 0.10 0.10 0.15 0.15 0.12 0.12 0.10 0.10 0.12 0.10 0.10

Dismantling Cost ($) 0.15 0.15 0.10 0.10 0.05 0.05 0.10 0.10 0.10 0.10 0.05 0.05 0.06 0.05 0.05

Components are named from A to O. The convention is that operative tasks are noted in the same way: A means the disassembly of the LCD monitor.

Table 3. End of life values and utilization rates


Reusable value ($) 60[65%] 50[61%] 35[69%] 28[64%] 30[61%] 30[65%] 25[65%] 35[58%] 16[70%] 32[69%] 16[69%] 6[64%] 6[73%] 25[61%] 15[65%]

Recyclable value ($) 50[75%] 50[75%] 75[85%] 65[85%] 180[90%] 110[80%] 55[75%] 55[75%] 25[70%] 50[70%] 50[80%] 50[80%] 15[65%] 30[75%] 25[70%]

With these values as initial data and taken into account the assumptions made in constructing the model, we performed the simulation. 5.2

BayesiaLab

BayesiaLab [15] is a tool for graphical manipulation of Bayesian networks. It allows defining, modifying, using and learning models based on Bayesian networks. It is also a decision aid instrument since it allows introducing decision and utility nodes. Therefore, the best decision can be found at the end of the inference process induced by simulation and the also the utility of this decision in terms of profit. The qualitative data in Bayesialab is represented by the structure of the BN graph (Nodes and Arcs), and the quantitative information by the Conditional Probability Table and Databases. It is possible to learn a Bayesian network considering an initial structure. To take into account this a priori knowledge, a virtual database with N samples (where N corresponds to the number of cases that have been used to set this a priori knowledge) must be provided. The distribution of these samples corresponds to the joint probability distribution represented by the initial Bayesian network. This virtual database and the real database are then both taken into account by the learning algorithms to induce a new Bayesian network. In addition to the unquestionable observations, BayesiaLab is also able to seize degrees of probability. Once validated, probabilities are used jointly with the probability distribution for giving a new Probability distribution. BayesiaLab allows the temporal dimension integration in a Bayesian Network.

A BN can be easily transformed into a DBN. Temporal nodes at instants t and t+1 can be represented and connected by temporal arcs. The parameters evolution of the DBN nodes can be so followed in time. Concerning the way of reasoning of a DBN, different kind of inference algorithms are available. Among those algorithms, particularly popular is the junction tree (JT) inference. It is based on the construction of a classical BN inference data structure called junction or join tree and belonging to the category of exact inference algorithms. Another inference algorithm is the Boyen–Koller (BK) algorithm, a parameterized procedure that, depending on the parameters provided may return exact as well as approximate results [16]. 5.3

Implementation

Launching the simulation one could observe the learning process of the DBN. Using a learning algorithm the missing data is dynamically completed and the joint probabilities are updated at each iteration execution (Fig 2). One can observe that end of life options, costs and revenues influence the type of disassembly operation so as the decision is taken dynamically considering the state of the component. The main variables and their values are dynamically represented using histograms (Fig 3).

Fig. 2. Completion of variables values (revenue)

Fig. 3. Dynamical representation of variables and decision making during the simulation

Evolution of main indicators can be followed on the temporal graph. Here costs and revenues variability and decision utility are represented.

Fig. 4. Temporal graph for variables dynamic

Disassembly operations are fulfilled in 45% of cases, the rest of 65% being destructive operations. More than 75% of operative costs have high values while most of the revenues are situated under 120 unities. This result shows that the disassembly operation is expensive. Sometimes dismantling operations are preferred instead. In this case we deal with notebooks which components are them selves expensive so one prefers to disassemble all its possible so to obtain the maximal profit. The minimal value of the decision utility is 17% which is also the value of the minimal profit. Decision values are given in the Fig 5. One can notice that components with high end of life values (such as monitors, network cards or keyboards) have to be disassembled. In the same time, most of the components suffer dismantling operations (such as motherboards, processors and batteries)

Fig. 5. Decision values

6

Conclusions

This paper proposes a method to establish the influences and the causalities between the main disassembly performance indicators in order to decide the optimal disassembly policy. A Dynamic Bayesian Network model is developed to highlight the temporal dependencies and causalities between variables of the system. The dynamic behavior of the disassembly process is well described by its stochastic variables: operative times, operative costs, type of operation, components end of life states. The proposed model is then validated using the BayesiaLab software. The method is implemented and validated on a reference case study from literature. The proposed model is useful in real time decision making process planning when one has to decide the type and the depth of disassembly operation. However, the model doesn’t take into account other end of life options (as discarding, melting, grinding, etc) as well as different variants of the product.

References 1. Pearce, I. A. (2009) The profit making allure of product reconstruction. MIT Sloan Management Review 50(3), 59-63. 2. Lambert, A. J. D. and Gupta, S. M. (2005) Disassembly Modeling for Assembly, Maintenance, Reuse, and Recycling, CRC Press, Boca Raton, Florida 3. Geiger D., Zussmann E. (1996) Probabilistic Reactive Disasssembly Planning, CIRP Annals, 1996, vol. 45, no1, pp. 49-52 4. Grochowski D. E., Parameter Estimation for Optimal Disassembly Planning,IEEE, pp 24902496, 2007 5. Bayındıra Z.P., Dekkerb R, Porrasb E., 2006, Determination of recovery effort for a probabilistic recovery system under various inventory control policies, The International Journal of Management Science, Vol 34, pp. 571 – 584. 6. Jensen F. V. Bayesien Networks and Decision Graphs, Springer-Verlag, 2001. 7. Clemen R, Reilly T,. Making hard decisions with Decision Tools . Duxbury Thomson Learning, 2001 8. Godichaud M., 2010, Outils d’aide à la décision pour la sélection des filières de revalorisation des produits issus de la déconstruction des systèmes en fin de vie, thèse de doctorat, Université de Toulouse. 9. Alami, M., 2009, Lot économique de pièces de rechange à produire en tenant compte des différentes phases du cycle de vie du produit, Thèse de doctorat, Université Laval, Québec. 10. Conrady S, 2011, Introduction to Bayesian Networks Conrady Applied Science, LLC Bayesia’s North American Partner for Sales and Consulting 11. Hu J., Zhang L., Ma L, Liang W., 2011, An integrated safety prognosis model for complex system based on dynamic Bayesian network and ant colony algorithm, Expert Systems with Applications 38, pp. 1431–1446 12. Portinale L., Raiteri D.C., Montani S., 2010, Supporting reliability engineers in exploiting the power of Dynamic Bayesian Networks, International Journal of Approximate Reasoning 51, pp. 79–195. 13. Duta L., Caciula I, Addouche S. On the profitability of the disassembly processes, accepted at The 18th IFAC World Congress, Milan 2011 14. Imtanavanich, P. and Gupta, S. M. (2007) Generating a Disassembly-to-Order Plan,

Proceedings of the 2007 Northeast Decision Science, on CD-ROM, 2007 15. http://www.bayesia.com/en/products/bayesialab.php 16. Murphy K. P, 2002, Dynamic Bayesian Networks: Representation, Inference and Learning, Thesis University of California.