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A Stochastic Petri Net Approach for the Manufacturing System Design DANIELA COMAN, ADELA IONESCU, MARIUS GIGI COMAN Department of Engineering University of Craiova Calugareni 1 street, Code 220037 ROMANIA
[email protected] http://www.imst.ro Abstract: - The design of a manufacturing system requires modeling and performance evaluation techniques. This paper is focused to a stochastic Petri net approach for the modeling and simulation of a manufacturing system. The stochastic Petri net model is implemented in Petri Net Toolbox under MATLAB environment. It is achieved the graphic construction of the net. Therefore, it is validated the net topology, the evolution of (their dynamics), as
well as the structural and behavioral properties (corresponding to checking if resources usage is stable and the model have no deadlocks). Some global performance indicators are determined in order to evaluate the performance of the manufacturing system. Key-Words: - manufacturing system, stochastic Petri nets, performance indicators Petri net. In section 4, the stochastic Petri net model are implemented in MATLAB environment, being obtained the global performance indicators. Concluding remarks follow in section 5.
1 Introduction Today's manufacturing systems are highly complex and many are very costly to build and maintain. A manufacturing system is a technical system in itself and it is defined to be a collection of integrated equipment and human resources, whose function is to perform one or more processing and/or assembly operations on a starting raw material, part, or set of parts. The integrated equipment includes production machines and tools, material handling and work positioning devices, and computer systems to coordinate and/or control the preceding equipments. Among the methods applied to the flexible manufacturing systems the Petri net approach seems to be a very promising one. Petri nets allow one to model many structural and behavioural features typical for the manufacturing systems. Using Petri net models it is possible to includes the hierarchical structure of the manufacturing system, different material and data flows, concurrency and asynchronicity of process execution, buffer sizes, machining operation times, and so on. In this paper we present an overview of the use of simulation in the design and analysis of a manufacturing system. Petri net based distributed system modelling takes place at the state level: it determines the actions that take place in the system, the states that precede these actions and the state in which the system will pass after the actions have taken place. [2] By simulating the state model through the Petri nets, we obtain a description of the system’s behaviour. [1] In section 2, the basic theory of Petri and stochastic Petri nets it is presented. In section 3, it is illustrated how a manufacturing system is designed using stochastic
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2 Petri Nets and Stochastic Petri Nets Petri nets, developed by Carl Adam Petri in his Ph.D. thesis in 1962, are generally considered as a tool for studying causality and inferencing relations. Formally, the structure of a Petri net is defined by its places, transitions, input function and output function. [4], [8] Definition 2.1 (Petri Net Structure): A Petri net structure, C, is a four-tuple, C = (P, T, I, O), where: P = {p 1 , p2 , …, p n } is a finite set of places, n ≥ 1; T = {t1 , t2 , …, tm } is a finite set of transitions, m ≥ 1; I : T → P∞ is the input function, a maping from transitions to bags of places, O : T → P∞ is the output function, a mapping from transitions to bags of places. Definition 2.2 (Petri net graph): A Petri net graph G is a bipartite directed multigraph, G = (V, A), where V = {v1 , v2 , . . ., vs } is a set of vertices and A = {a 1 , a 2 , . . . , a r } is a bag of directed arcs, a i = (vj , vk), with vj , vk V. The set V can be partitioned into two disjoint sets P and T such that V = P T, P T = , and for each directed arc, ai A, if a i = (vj , vk), then either vj P and vk T or vj T and vk P. A marking µ is an assignment of tokens to the places of a Petri net. A token is a primitive concept for Petri nets, like places and transitions are. Tokens are assigned to, and can be thought to reside in, the places of a Petri net. The number of tokens and the places they reside in
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may change during the execution of a Petri net. The tokens are used to define the execution of a Petri net. Definition 3 (Marking): A marking µ of a Petri net C = (P, T, I, O) is a function from the set of places P to the nonnegative integers N, µ : P → N. It can also be defined as an n-vector, n = | P |, such that µ = (µ(p 1 ), µ(p 2 ), ..., µ(p i), ..., µ(pn )). µ(p i ) gives the number of tokens in place p i . Definition 4 (Marked Petri): A marked Petri net M = (C, µ) is a Petri net structure C = (P, T, I, O) and a marking µ. A transition is enabled if each input place contains at least one token; an enabled transition fires by removing a token from each input place and depositing a token in each output place. Stochastic Petri net models were proposed by researchers active in the applied stochastic modeling field, with the goal of developing a tool which allowed the integration of formal description, proof of correctness, and performance evaluation. This type of Petri nets are special extensions of timed transitions Petri nets, in which the transition time is not deterministic. This extension augments the modeling power, allowing systems that are affected by nondeterministic factors (in our case human behavior) to be modeled. The transition firing time is usually described by a probabilistic distribution, and commonly exponential distributions are used. The formal definition of stochastic Petri net (P,T, I, O, M 0 , ) is given bellow [3], [5]: 1. P = {p1 , p 2, ..., pn }, n >0 : finite places set; 2. T = {t1 , t2 , …, tm } m > 0: finite transitions set, given P T ≠ and P T = ; 3. I : P x T → N is the input function which defines the set of arcs directed from P to T where N = {0, 1, 2, ...}; 4. O : P x T → N is the output function which defines the set of arcs directed from T to P; 5. M : P → N : marking in which for every place p P there are markings. The initial marking is denoted M 0 ; 6. : T → R+ firing function in which i is the firing rate of the transition ti . The firing of a transition is an atomic operation, i.e., tokens are removed from its input places and deposited into its output places with one indivisible operation. A firing delay is associated with each transition. It specifies the amount of time that must elapse before the transition can fire. This firing delay is a random variable with negative exponential probability density function. The parameter of the probability density function associated with transition ti is the firing rate associated with ti , i . This firing rate may be marking-dependent, so that it should be written i (Mj ).
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The average firing delay of transition ti in marking M j is [i (M j )]-1 .
3 Manufacturing System Design The manufacturing system for the present study includes four machines (M1, M2, M3 and M4) capable of performing a variety of tasks; an automated guided vehicle based material handling system and a single input, single output storage-retrieval system connected to the manufacturing system by conveyors. Three types of parts (P1, P2, P3) are processed on each machine. The processing of a part can start as soon as a part of the same type has been completed and the required machine is available. It is assumed that the number of raw parts is sufficiently high. The times for transport operations and for the resources release are considered negligible. The following places and transitions are defined for the modeling of the manufacturing system in the Petri net:
Structure: (P, T, I, O). Places: P = {p1 , p2 , p3 , p4 , p5 , p6 , p7 , p8 , p9 , p10 , p11 , p12 , p13 , p14 , p15 , p16 , p17 , p18 , p19 , p20 , p21 , p22 , p23, p24 }. Transitions: T = {t 1 , t 2 , t3 , t4 , t5 , t 6 , t 7 , t8 , t9 , t10 , t11 , t 12 , t13 , t14 , t15}. Input function: I(t 1 ) = {p10 }, I(t 2 ) = {p1 , p12 }, I(t 3 ) = { p2 , p14}, I(t 4 ) = {p3 , p22}, I(t 5 ) = {p4 }, I(t 6 ) = {p5 , p11 }, I(t 7 ) = {p6 , p13}, I(t 8 ) = {p15 , p20 }, I(t 9 ) = {p16 }, I(t10 ) = {p7 , p17}, I(t 11 ) = {p8 , p18 }, I(t12 ) ={p9 , p24 }, I(t 13 ) = {p19}, I(t 14 ) = {p23}, I(t 15 ) = {p21}. Output function: O(t 1 ) = {p1 }, O(t 2 ) = {p2 , p11}, O(t 3 )={p3 , p13}, O(t 4 ) = {p15 , p19}, O(t 5 ) = {p16 }, O(t 6 ) = {p4 , p17 }, O(t 7 )= {p5 , p18}, O(t 8 ) = {p6 , p24 }, O(t 9 ) = {p7 }, O(t 10 ) = {p8 , p10 }, O(t 11 )={p9 , p12 },O(t12 )={p14 , p21 }, O(t 13 ) = {p23}, O(t 14 ) = {p20 }, O(t 15 ) = {p22}. The Petri net model is presented in Fig. 1.
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Fig. 1.
4 Simulation Studies The simulation studies are carried out for the Petri net giving statistics on events and resources for model [6], [7]. The simulation of the proposed manufacturing system using the stochastic Petri nets provides the possibility to view the manufacturing process in time. After achieving the graphic construction of the net, transporting it into a specific mathematical formalism has been made, so that the fulfiled structure to be fully retrieved and used to bring out the internal dynamics of the model. By the simulations using Petri Net Toolbox in Matlab environment, has been validated the net topology, the evolution of (their dynamics), as well as structural and behavioral properties (Fig. 2 - Study of the structural properties).
Fig. 2.
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In the Petri Net Toolbox, by the simulation of the system functioning for a period of 1000 hours have been obtained global performance indicators. These indicators, presented in the following three tables (Table 1 and Table 2 - Global Statistics Places and Table 3 - Global Statistics Transitions) demonstrate the performance of the considered manufacturing system. Table 1 Place Name
Arrival Sum
Arrival Rate
Arrival Dist.
Throughput Sum
p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23 p24
4563 4563 4563 4563 4563 4563 4562 4562 4562 4562 4563 4562 4563 4562 4563 4562 4563 4563 4563 4563 4562 4562 4563 4563
0.07605 0.07605 0.07605 0.07605 0.07605 0.07605 0.076033 0.076033 0.076033 0.076033 0.07605 0.076033 0.07605 0.076033 0.07605 0.076033 0.07605 0.07605 0.07605 0.07605 0.076033 0.076033 0.07605 0.07605
13.1493 13.1493 13.1493 13.1493 13.1493 13.1493 13.1521 13.1521 13.1521 13.1521 13.1493 13.1521 13.1493 13.1521 13.1493 13.1521 13.1493 13.1493 13.1493 13.1493 13.1521 13.1521 13.1493 13.1493
4563 4563 4563 4562 4563 4563 4562 4562 4562 4563 4563 4563 4563 4563 4563 4562 4562 4562 4563 4563 4562 4563 4563 4562
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Table 2 Place Name
Throughput Rate
Throughput Dist.
Waiting Time
Queue Length
p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23 p24
0.07605 0.07605 0.07605 0.076033 0.07605 0.07605 0.076033 0.076033 0.076033 0.07605 0.07605 0.07605 0.07605 0.07605 0.07605 0.076033 0.076033 0.076033 0.07605 0.07605 0.076033 0.07605 0.07605 0.076033
13.1493 13.1493 13.1493 13.1521 13.1493 13.1493 13.1521 13.1521 13.1521 13.1493 13.1493 13.1493 13.1493 13.1493 13.1493 13.1521 13.1521 13.1521 13.1493 13.1493 13.1521 13.1493 13.1493 13.1521
1.4922 1.3595 1.2859 0.995 1.0065 1.0116 1.0031 1.0134 0.99113 1.017 7.6415 1.496 5.2755 1.8646 2.978 1.0011 2.9993 5.0193 0.9891 1.0038 1.0058 2.1449 0.98507 7.0223
0.11348 0.10339 0.097791 0.075653 0.076541 0.076934 0.076272 0.077053 0.075359 0.077341 0.58113 0.11377 0.4012 0.1418 0.22648 0.076119 0.22804 0.38164 0.075221 0.076342 0.076472 0.16312 0.074915 0.53393
index associated to the t14 transition, Figure 7 Dependency of the Utilization index associated to the t15 transition, Figure 8 - Dependency of the Waiting Time index associated to the p18 position, Figure 9 Dependency of the Waiting Time index associated to the p10 position, Figure 10 - Dependency of the Waiting Time index associated to the p6 position, Figure 11 Dependency of the Queue length index associated to the p1 position, Figure 12 - Dependency of the Queue length index associated to the p9 position) , Figure 13 Dependency of the Queue length index associated to the p24 position).
Fig. 3
Table 3 Transition Name
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15
Service Sum 4563 4563 4563 4563 4562 4563 4563 4563 4562 4562 4562 4562 4563 4563 4562
Service Rate 0.07605 0.07605 0.07605 0.07605 0.076033 0.07605 0.07605 0.07605 0.076033 0.076033 0.076033 0.076033 0.07605 0.07605 0.076033
Service Dist. 13.1493 13.1493 13.1493 13.1493 13.1521 13.1493 13.1493 13.1493 13.1521 13.1521 13.1521 13.1521 13.1493 13.1493 13.1521
Service Time 0.56426 0.68083 0.84037 0.97183 0.995 1.0065 1.0116 1.0038 1.0011 1.0031 0.76252 0.67876 0.9891 0.98507 0.65645
Utilization 0.042912 0.051777 0.06391 0.073908 0.075653 0.076541 0.076934 0.076342 0.076119 0.076272 0.057977 0.051608 0.075221 0.074915 0.049912
Fig. 4
Also, the special options of Petri Net Toolbox, which confers a high capacity of analysis, has made possible a synthesis of this Petri net model which allows exploring the dependences of global performance indicators associated with the net positions/transitions on two “Design Parameters” (being considered transitions t1 and t12) for the various parameters of the simulation (Figure 3 - Dependency of the Service Sum index associated to the t3 transition, Figure 4 - Dependency of the Service Rate index associated to the t1 transition, Figure 5 – Dependency of the Service Dist. index associated to the t2 transition, Figure 6 - Dependency of the Service Time
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Fig. 5
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Fig. 6
Fig. 9
Fig. 7
Fig. 10
Fig. 8
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Fig. 11
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IMACS World Congress Scientific Computation, Applied Mathematics and Simulation, 2005. [3] P. J. Hass, Stochastic Petri Nets for Modeling and Simulation, Winter Simulation Conference, 2004, pp. 101-112. [4] T. Murata, Petri nets: Properties, analysis and applications, Proceedings of the IEEE, 1989, pp. 541 – 580. [5] J. A. Buzacott, J. G. Shanthikumar, Stochastic models of manufacturing systems, Prentice Hall, 1993. [6] O. Pastravanu, M. Matcovschi, C. Mahulea, Aplications of Petri Nets in discreet systems analysis, “Gh. Asachi” University Press, 2002. [7] O. Pastravanu, Discreet systems analysis. Qualitative techniques based on Petri net formalism, MatrixRom Press, 1997. [8] J. L. Peterson J. L., Petri net theory and modelling of systems, Prentice-Hall Inc., 1981.
Fig. 12
Fig. 13
5 Conclusion In conclusion, it can be said that stochastic Petri nets are a simple but effective method of analysing manufacturing systems. The use of the stochastic Petri nets will surely lead to more efficient and stable manufacturing systems being implemented and therefore increasing the productivity and efficiency of modern manufacturing methods.
References: [1] J. Fowler, A. Schomig, Applied system simulation: methodologies and applications, Kluer Academic Publishers, 2003. [2] E. J. Lee, N. Dangoumau, A. Toguyeni, A Petri net based approach to design controllers for Flexible Manufacturing Systems, Proceedings of the 17th
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