MRP SYSTEMS APPLIED TO pt-temporized PETRI NETS

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1 MRP SYSTEMS APPLIED TO pt-temporized PETRI NETS Carlos Augusto de Alcantara Gomes Adjunct Professor of Industrial Engineering Department at Universidade Federal do Rio de Janeiro Address: Rua Almte. Cóchrane 56 apt Tijuca - Rio de Janeiro - RJ - BRAZIL, CEP ABSTRACT Petri Nets for Production Systems are shown in this paper as pt-temporized Petri Nets, with its initial marking having only one token in the first place. pt-temporized Petri Nets are an easy way to control production systems, because their performance, can be seen in a real-time and their hierarchizing evidence the consumption of resources (workmanship, equipments, materials, etc.). MRP Systems are used for deciding the material request for Flexible Manufacturing Industries. Keywords: Petri Nets, Manufacturining Requirement Planning, Civil Production 1. INTRODUCTION Through a Potential-Task graph we can associate to each node a transition and to each arc a place that interconnects two transitions so, in that way, a Petri Net is equivalent to the following graph. (ALCANTARA GOMES, 1990). A 4 D 0 3 I 4 F 0 6 B C - task - duration Fig. 1 - Potential-Task graph

2 p 4 p 7 t D p 2 t A t F p 9 p 1 t I p 5 p 3 t C p 8 p 6 t B - place - transition - oriented line-connectivity place-transition and vise-versa - token Fig. 2 - An associated Petri Net In the case of a Potential-Task graph, the studied resources in the production process do not appear explicitly being only described by terms of immersed ways. Concerning to Petri Nets resources can be visualized easily which are assigned to tokens. node (task) transition arc place resource token 2. pt-temporized PETRI NETS ASSOCIATED TO THE PRODUCTION SYSTEMS IN FLEXIBLE MANUFAC-TURING SYSTEMS pt-temporized Petri Nets are Petri Nets in which the time variable is associated to places and transitions, so this way you define: pt-tpn = {P, T, Pre, Pos, M o, d, δ} Where: P = (p 1, p 2,..., p m ) is a set of places T = {t 1, t 2,...,t n } is a set of transitions Pre: P T N is a input function of transitions (N is a set of nonnegative integers) Pos: T P N is a output function of transitions M o : P N is a function from a set of places P to the nonnegative integers and is defined by {m 1, m 2,...,m m }. Where m 1 = 1 and m 2, m 3,..., = 0, what is the initial marking d = {d 1, d 2,... d m } is the finite set of places duration δ = {δ 1, δ 2,...,δ n } is the finite set of transitions duration

3 [ d 4 ] [ δ 4 ] [ d 7 ] p 4 p 7 [ d 2 ] [ δ 2 ] t 4 p 2 [ δ 6 ] [ d 8 ] [ d 1 ] [ δ 1 ] t 2 [ d 5 ] t 6 p p 5 p 1 t 1 [ d 3 ] [ δ 5 ] [ d 8 ] p 3 [ δ 3 ] [ d 6 ] t 5 p 8 t 3 p 6 Fig.3 - pt-temporized Petri Net The pt-temporized Petri Nets can represent a production system and can also control it in a real time manner. In this way, whenever a transition fires an activity is accomplished and firing a transition it removes the token(s) from its input place that causes the consumption of resources that can be easily controlled by the performance of the net. pt-temporized Petri Nets permit the hierarchizing, that is, each place or transition can be separated into another Temporized Petri Net. In this case, there is a difference in the marking, because it is no more given by the form M = {1, 0, 0,..., 0} but the marking M that associates to each place a number of tokens M(p) 0. In the figure 4 it is shown that a transition t 1 can be separated in another Temporized Petri Net, and the original token that leaves p 11 and the tokens that leave p 12 are the resources used for the firing of transition t SYSTEMS OF PRODUCTION CONTROL WITH pt-temporized PETRI NETS The execution of a pt-temporized Petri Nets is controlled by the number and distribution of tokens in the Petri net. Tokens reside in places and control the execution of the transitions of the net. A pt-temporized Petri Net executes by firing transitions. A transition fire by removing tokens from its input places and creating new tokens that are distributed to its output places. A transition may fire if it is enable. A transition is enabled if each of input places has at least as many tokens in it as arcs from the place to the transition. Multiple tokens are needed for multiple input arcs. The tokens in the input places that enable a transition are its enabling tokens. A transition fires by removing all of its enabling tokens from its input places and then depositing into each of its output places one token for each arc from the transition to place. Multiple tokens are produced for multiple output arcs. Firing a transition will in general change the marking M of the pt-temporized Petri Nets to a new marking, M. Notice that since only enabled transitions may fire, the number of tokens in each place always remains nonnegative when a transition is fired. Firing a transition can never try to remove a token that is not there. If there are not enough tokens in any input place of a transition, then the transition is not enable and can not fire

4 [ δ 1 ] t 1 [ d 12 ] [ δ 12 ] p 12 t 12 [ d 11 ] [ δ 11 ] [ d 14 ] p 11 t 11 [ d 13 ] [ δ 13 ] p 14 p 13 t 13 Marking M o = {1, 2, 0, 0} Fig.4 - Temporized Petri Net for a higher level of Hierarchy A transition t in a marked pt-temporized Petri Net with marking M may fire whenever it is enabled. Firing an enabled transition t results in a new marking M. For the pt-temporized Petri Net there is a need to know the date of the earliest and the latest fire of a transition, the date of the entrance of the earliest and the latest token in a place, besides that you must know the longest path which goes from the source of the net to its sink. This is the path that must get the greatest control, because a delay causes another delay at the net duration. The knowledge of these dates and this path are very important to control the pt- Temporized Petri Net in a possible real-time. What implies knowing which transition (task) is enable to fire and which tokens (resources) are used. Therefore the sequence of transition firing is obtained (see ALCANTARA GOMES, 1990), which permits to see how the nets evaluate during the manufacture of the product.

5 4. ALGORITHM FOR DETERMINING THE EARLIEST AND THE LATEST DATE OF A TRANSITION FIRING, THE EARLIEST AND THE LATEST DATE OF ENTRANCE OF A TOKEN IN THE PLACE AND THE LONGEST WAY WHICH GOES FROM THE SOURCE TO THE SINK IN pt- TEMPORIZED PETRI NET. (1) Sort a pt-temporized Petri Net according to the levels of the places and the transitions appear on the net. Even levels are associated to transitions and odd levels to places. The smallest index of a place (transition) of a given odd (even) level can not be higher than the lower index of a place (transition of posterior level), see figure 5. (2) A ruled in squares is formed and divided horizontally and vertically, in a number of parts equivalent to the addition of the number of places and transitions. On the above horizontal part and on the left vertical the places and the transitions are written according to the level and as the figure 6 indicates. (3) On the diagonal are assigned the duration of the places and the transitions respectively on the horizontal part below, an inferior row of squares receives the dates from the earliest fire of the transitions and the earliest date of entrance token in the places. A right column of small receives the dates from the latest transitions firing and from the latest dates of entrance tokens in the place. (4) The adjacency matrix is used in a ruled of squares where p k -t n is a way that has its provenance in the place p k and its destination in the transition t n and vice versa, in which the number 1 will be assigned. (5) The date of the earliest fire transition and the earliest date of entrance token on the place is calculated at first. The value of the first square from inferior row of squares is zero. Transplaces this value up vertically to the main diagonal and add it to the value there. Then transplace this sum horizontally to meet a value 1 from the adjacency matrix, go down vertically and assign it in a square of inferior squares. The process must be repeated to all values 1 of the adjacency matrix when the sun is moved to the right. This process is repeated as many times as the valued squares of interior row of squares. When more than a value 1 in the work column or more than a value on the square of inferior square is met the highest number must be chosen, see figure 6. (6) Determination of the latest fire date from a transition and the latest date of entrance place token. It is allowed that the latest date of the entrance in the last place is equal to the earliest date of entrance of the last place therefore this value is assigned in the last square of the right column of squares. Then go horizontally, from the above mentioned value, until the main diagonal. From this point go vertically up until the value 1 is met. The value found in the main diagonal placed in the same horizontal line is the value that will be subtracted from the above mentioned value. The resulting value is placed in the same horizontal line on the right column of squares. This process is repeated as many times as the valued squares of right column of squares. If more than a value 1 is met the process must be repeated until all these values are treated. In the case of adjacency matrix gets in

6 the same line more than one value 1 it will be evaluated on the right column of squares and the smallest one must be chosen, see figure 6. (7) The longest path that crosses the smallest slacks (slack is the difference between the latest date and earliest date of the fire transition, or the smallest entrance of a token in the place and the earliest one) is the path that goes from the source to the sink and this path must be the most controlled, see figure 5. [ 2 ] [ 0] [ 1] p 2 ( 11, ) t 2 ( 3, 3) p 4 ( 11, ) [ 3] p t 1 ( 0, 0) [ 3] p 3 ( 11, ) [ 1] t p 5 ( 5, 5) t 5 ( 7, 7) [ 0] p 7 ( 10, 10) [ 1] t f = 1 p 6 ( 6, 7) f = 1 t f = Fig. 5 - pt-temporized Petri Net separated by level The longest paths that go from the source to the sink are obtained by the step (7) of the algorithm and they are: p 1 - t 1 - p 2 - t 2 - p 4 - t 5 - p 7 p 1 - t 1 - p 3 - t 3 - p 5 - t 5 - p 7 and p 1 t 1 p 2 p 3 t 2 t 3 t 4 p 4 p 5 p 6 t 5 t 6 p 7 p t ;0 p p ;1 t t t p p p t t p Fig. 6 - A ruled of squares for determining the earliest and the latest date of transition fire, the earliest and latest date of entrance of token in the place and longest path which goes

7 from the source to the sink in a pt-temporized Petri Net 5. pt-temporized PETRI NETS ASSOCIATED TO MRP SYSTEMS IN FLEXIBLE MANUFACTURING SYSTEMS The association of Petri Nets to MRP Systems are done using the earliest fire of the transitions for determining request dates and the necessary resources at the transitions to determine the lot size. Then following with the data presented below, we will show the MRP System applied to cement bags requests, at a civil construction (that is a kind of Flexible Manufacturing System). The way considered at the example is p 1 - t 1 - p 3 - t 3 - p 5 - t 5 - p 7, and the lead time is three weeks. Transition EDTf (week) Necessities (bags) Received (bags) 300 Stock (bags ) Lot 1 Request (bags) Lot are 300 bags of cement; - Lead Time is 3 (three) weeks; - EDTf = earliest date of transition fire. BIBLIOGRAPHY ALCANTARA GOMES, C. A. (1990) Contribuição a Modelagem e Análise de Sistemas Utilizando Redes de Petri: Com Aplicação à Indústria da Construção Civil D.Sc. Thesis, COPPE/UFRJ, Rio de Janeiro, Brazil CHRETIENNE, P. (1983) Les Réseaux de Petri Temporisés Tése, Université de Paris IV, Paris, France. PETERSON, J. L. (1981) Petri Net Theory and Modeling of Systems," Printece Hall, U.S.A., p.290 ALCANTARA GOMES, C. A. and QUALHARINI, E. L. (1996) Algorithm for Determination of The viable states, in a pt-temporized Petri: Net, Associated to Production Systems Latin-Iberian-American Congresss on Operations Research and Systems Engineering. pp , Rio de Janeiro, Brazil FULLMANN, C.; RITZMAN, L.; MACHADO, M et al (1989) MRP - MRPII - OPT - GDR. IMAN, Brazil. MONKS, J. G. (1987), Administração da Produção. McGraw-Hill.

DES. 4. Petri Nets. Introduction. Different Classes of Petri Net. Petri net properties. Analysis of Petri net models

DES. 4. Petri Nets. Introduction. Different Classes of Petri Net. Petri net properties. Analysis of Petri net models 4. Petri Nets Introduction Different Classes of Petri Net Petri net properties Analysis of Petri net models 1 Petri Nets C.A Petri, TU Darmstadt, 1962 A mathematical and graphical modeling method. Describe