## Abstract

To realize effective reuse of mechanical parts for the development of a sustainable society, it is essential to manage individual parts over their entire life cycle. Product users have difficulties carrying out appropriate maintenance on the multitude and variety of parts in their products. Addressing these considerations, we propose a scheme whereby a part manages itself and supports user maintenance activities. In previous work, we proposed and developed an application of Bayesian estimation to a part agent system that advises a user regarding the replacement of hard disk drives (HDDs). In this study, we create a Bayesian network on the deterioration of the HDD to find the probability of an unobservable event. We also discuss the application of this method to life cycle simulation performed by part agents.

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## References

Ministry of the Environment. http://www.env.go.jp/recycle/circul/reuse/index.html. Accessed 10 June 2017.

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Ueno T, Katou K, Ookawa H, Nanjyo K, Kawaharada H, Hiraoka H. Reuse support of mechanical parts by part agents using bayesian estimation. J JSPE. 2013;79(12):1258â€“64. (in Japanese).

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Nanjo K, Yamamori Y, Yokoki Y, Sakamoto Y, Hiraoka H. Maintenance decisions of part agent based on failure probability of a part using Bayesian estimation. In: The 22nd CIRP conference on Life Cycle Engineering, Sydney, Australia; 2015.

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## Acknowledgment

This work was supported by JSPS KAKENHI Grant Number 15Â K05772.

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## Appendix Bayesian Network

### Appendix Bayesian Network

Predicting failures of a part is an important function of the part agent to support the effective reuse of the part. However, it is also a difficult issue due to its probabilistic nature caused by its dependency on the level of usage by the consumer and on environmental conditions. To deal with this problem, we have applied an estimation method based on a Bayesian inference.

Consider the causal relation exists between events *A* and *B* where event *A* affects event *B*. The probability of the occurrence of an event *A* before the occurrence of related events is called the prior probability of *A* and is denoted by *P*(*A*). The probability of the occurrence of an event *B* after the occurrence of event *A* is called the conditional probability and is denoted by *P*(*B*|*A*). If we know the prior probability *P*(*A*), *P*(*B*), and the conditional probability *P*(*B*|*A*), we can estimate *P*(*A*|*B*), namely, the probability of the occurrence of *A* when you know event *B* occurred, by Bayes theorem
represented by the following Eq. (3.1):

This probability is called the posterior probability . It is an improved estimation of the probability of the occurrence of an event based on the knowledge of the occurrence of related events.

When multiple events affect an event, the causal relations among the events form a network. It is called a Bayesian network that is represented by a noncircular directed graph where each node is equipped with a table representing the conditional probabilities of related events called the conditional probability table.

Figure 3.11 shows a simple example of a Bayesian network. The graph shown in the left of the figure depicts that the probabilities of events A and C affect the occurrence of event B. The conditional probability of an event B or P(B|A,C) varies with the occurrence (shown as 0 and 1 in the table) of events A and C, which is summarized in the conditional probability table shown in the right of the figure

In a Bayesian network, there are three types of events that are input events, observable events, and unobservable events. The probability of the occurrence of unobservable events in the network can be estimated using Bayes theorem described above if we have conditional probabilities of events and gather evidences on the occurrences of input events and observed events.

We represent probabilistic causal relationships between failures of a part and their factors using this Bayesian network in order to obtain the probability of the failures.

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Fukunaga, Y., Fukumashi, Y., Nagasawa, A., Hiraoka, H. (2019). Bayesian Estimation for the Reuse of Mechanical Parts Using Part Agents. In: Hu, A., Matsumoto, M., Kuo, T., Smith, S. (eds) Technologies and Eco-innovation towards Sustainability I. Springer, Singapore. https://doi.org/10.1007/978-981-13-1181-9_3

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