# Part flow planning in a manufacturing cell based on Petri nets simulation.

1. INTRODUCTION

The theory and tools for modelling discrete events systems, are, among others, min-max algebra, Markov chains, stochastic Petri nets, theory of queues and queuing networks. As generally accepted, the Petri nets theory and applications serve very well the demands of modelling the manufacturing systems, therefore this tool was used in the allegations of this paper. One of the software implementation of the Petri nets modelling paradigm used in this paper has been developed by Robin Milner and his group at Edinburg University, using the Standard ML language. The network built as a model of a manufacturing system in the paper was run using a commercial computer program written for Unix (CPN Tools), as a performant implementation of Coloured Petri Nets (The CPN Group, 2009).

Subjective probability, first introduced as a concept by Bayes and referenced in many further papers as (Hietikko et al., 1990) and applied in many research as shown in (Hietikko, 1996), comes to offer more facilities in the process of decision making, giving the possibility to the controller of the system to decide the optimal continuation for routing the parts. In cooperation with the simulation tool, the Bayes's theory of subjective probabilities can generate an automated control system for the manufacturing cell (Bretthorst, 1988).

The CPN model of the manufacturing system, based on the layout in Fig. 1, is used for the simulation and detection of concurrency and resource sharing state of the system. Deadlock avoidance is based on a set of routing rules connected with the maximal values of posterior probabilities computed from the Bayes theorems.

2. THE CPN MODEL OF THE

MANUFACTURING SYSTEM

2.1 Layout of the manufacturing cell

The layout of the system is presented in Fig. 1.

[FIGURE 1 OMITTED]

The system is designed to produce both milled and turned parts, possibly with some welding operations. The material flow is mainly concentrated along the transfer line of the conveyor 1. There is also another conveyor, with a shorter displacement, which is intended to serve only the CNC lathe, in association with the robot. The milling center is placed at one end of the transfer line and it is served by conveyor 1 exclusively. The CNC milling machine has a two places input /output table. At the other end of the transfer line the system has the welding place with a rotating table. This place is served by the robot and conveyor 1. Two types of transfer pallets are used in the system, one for polyedrical parts as for milling operations and another one with parallel multiaxial display of parts, for turning operations. These two different types of pallets have different circuits in the cell and are stored separately in the two storage places.

2.2 The CPN model

The Coloured Petri Nets model (CPN) is built regarding the material flow in the system, starting from the two storages of pallets and having the end point either in the storage or at the human operator who can unload the part from the system.

The opening page of the model is presented in Fig. 2. The "Orders" place in the model is the place where production orders coming from the production process planning system (CAPP) are "placed". These orders can fire the transition representing the orders' submission, placing in this way other tokens (firing other transitions) in the subsequent places of the model. And so, the network runs in respect to the real behavior of the system. The opening page is only the "turntable" for the next components of the model, being able to "start" simultaneously different processes, like for instance milling of one part and welding of another. As far as the two processes use the same resource to transfer parts, which is the conveyor 1, these two processes cannot be served simultaneously in real time. This concurrency will determine a deadlock in the system.

In a real system the process planning designer is taking the decision based mainly on some common sense of the situation or is imposing severe and rigid exclusion rules for the conflict situation that might occur, such as coincident request for a resource. But these priority rules don't work in all situations. As mentioned in (Ruiz-Mier & Talavage, 1987), reasoning based on rigid rules can be confused by some unpredicted situation, like machine down, or misrouting of a part. Ruiz and Talavage (Ruiz-Mier & Talavage, 1987) were proposing the use of learning adjustable weights for different possible routes in the system, in order to optimize the material flow.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

An example taken from the simulation net of the system is emphasizing the situation when two transitions have concurrent conditions to fire, the "RawIn" and "PartOut". Fig. 3 is presenting the model in the concurrent state for conveyor 1 (position Convr_1).

3. CONTROL OF MANUFACTURING SYSTEM IN SUBJECTIVE PROBABILITY BACKGROUND

The decision rules defined on the model page are based on the concept of subjective probability (Hietikko, 1996) merged in the code involved in running the model, as well as in the code generated for controlling the real system. The most suspicious decision points in the system could be:

* selection of one loaded pallet from all available pallets in the storage;

* resuming the system after emergency stop;

* in cycle conveyor request by the human operator;

* conflicting request for conveyor 1 by multiple devices;

For every conflict point in the simulation there was added a set of criteria for reasoning:

* how far is the pallet from the transfer device;

* the level of importance for the operation served by the conveyor;

* distribution of parts in the system when resuming after emergency stop;

* the damages produced due to the emergency stop;

Some particular situations for routing the loaded pallets are simulated considering the values of subjective probabilities, representing the individual's confidence that the evidence E can be noticed when the proposition [z.sub.i] is known to be true and is denoted as P(E|[z.sub.i]).

If the proposition vector z = {[z.sub.1], [z.sub.2], [z.sub.n]} is including "n" exclusive propositions (for example the alternatives of decisions like Pallet #1 or Pallet #2) and there are multiple evidences E = {[E.sub.1], [E.sub.2], ..., [E.sub.m]} (for example the decision criteria like the importance of the operation), that are independent, the posterior probability for different propositions (optimal decision) can be calculated using equation (1) (Bretthorst, 1988).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Let us suppose that there are, for example, three pallets to choose using three decision criteria (subjective probabilities):

* the pallet near the conveyor must be selected in 80 % of cases;

* the conveyor must serve the most important machine in 90% of cases;

* the smallest lead time for a part will prevail in 70% of

cases;

The decision table with posterior probabilities (Tab. 1) is built using the subjective probability values. If, in the decision point, the conveyor is near pallet #1, the importance of pallets #2 and #3 are high and the lead time for pallet #2 is small, the decision will be, according to equation (1), for pallet #2.

4. CONCLUSIONS

As presented in the paper, on certain instances in the development of a production process sharing resources can determine deadlock states. The selection of the optimal transfer route is affected by a certain degree of uncertainty. When the simulation support is present, the uncertainty can be reduced by reasoning in a probability environment. In this way the optimal decision in the routing process can be reached considering a set of criteria in building the logic of the decision process based on Baye's theory.

For the future of this research the integration of the decision table in the control program of the manufacturing cell is to be considered. The optimization of the decision based on the occurrence graph (OG) and a procedure to select the deadlock risk for every state of the system must be developed. Another point of interest is to bring time in the simulation. In this way the model is more realistic and the simulation data can be more relevant for the development of the real system.

5. REFERENCES

Bretthorst, G. Larry, (1988), Bayesian Spectrum Analysis and Parameter Estimation, in Lecture Notes in Statistics, 48, ISBN 0-387-96871-7, Springer-Verlag, New York, New York

Hietikko, E., Lappalainen, P., Parkkinen, R., (1990), Diagnostic Control of an Automatic Production Line, Proceedings of Pacific Conference on Manufacturing, Australia: pg. 558-566

Hietikko E., (1996), Computer Aided System for Preliminary Design of Screen Cylinder Variants, Ph. D. disertation, Oulu Oulun Yliopisto, C90, 1996

Ruiz-Mier S., Talavage J., (1987), A Hybrid Paradigm for Modeling of Complex Systems, Artificial Intelligence, Simulation and Modeling, edited by Lawrence E. Widman, ISSN:0037-5497, A Willey-Interscience Publication: John Wiley & Sons

*** (2009) http://wiki.daimi.au.dk/cpntools/cpntools.wiki - The CPN Group, University of Aarhus, Denmark, Accesed on: 2009-05-03
```Tab. 1. Decision table with conditional probability values

Criteria/Decision pallet #1 pallet #2 pallet #3

Priors 0.5 0.5 0.5
Conveyor is near 0.8 0.8 0.8
Operation is important 0.9 0.9 0.9
Lead time is small 0.7 0.7 0.7
Posterior probabilities (Eq. 1) 0.12 0.62 0.26
```