**2. Petri Nets**

The PN describes the system structure as a directed graph and can capture precedence relations and structural links of real systems with graphical expressiveness to model conflicts and queues. Formally, it can be defined as a sixfold (P, T, A, M0, W, K) in wich: P is a set of states/places, T is a set of transitions, A is a set of arcs subject to the constraint that arcs do not connect directly two positions or transitions, M0 is the initial state, which tells how many marks/tokens there are in each position to the beginning of the processing, W is a set of arc weights, which tells, for each arc, how many marks are required for a place by the transition or how many are placed in a place after the respective transition; and K is a set of capacity constraints, which reports to each position, the maximum number of marks which may occupy the place (Castrucci; Moraes, 2001). Applying the definition in the PN of Figure 1, P = [p0, p1]; T = [t0]; A = [(p0, t0), (t0, p1)]; W: w (p0, t0) = 1, w (t0, p1) = 1 e M0 = [1; 0]. The token in p0 enables the transition t0. After firing, M = [0; 1].

The transitions correspond to changes of states and places correspond to state variables of the system. In the firing of a transition, the tokens move across the network in two phases: enabling and firing transition. A transition tj ∈ T is enabled by a token m if ∀ pi ∈ P, m (pi) ≥ w(pi, tj), i.e., the token in place pi is greater than or equal to the arc weight that connects pi to tj.

Some variations are allowed in Petri Nets and were used for modeling, for example, the use of inhibitor arcs.

**Figure 1.** Symbolic representation of Petri Nets

96 Petri Nets – Manufacturing and Computer Science

may occur in manufacturing.

**2. Petri Nets** 

of inhibitor arcs.

generate idle nor so high as to increase the throughput time.

which can be used in plans not yet released for manufacturing.

token in p0 enables the transition t0. After firing, M = [0; 1].

inventory level that will be allowed in manufacturing. This should not be so low as to

In the first two chapters will be presented basic concepts for modelling the proposed system using Petri Nets and throughput diagram, these methods will be applied in a real

The aim of this paper is to measure in advance in-process inventory and lead time in manufacturing that a production plan will generate. Knowing the magnitudes of the plan prior to release, a manager can predict and possibly prevent problems, changing the plan. The specific objectives were: i) mapping manufacturing, ii) model building for PN, refining and validated by field data, iii) with the results simulated by throughput diagram, calculate the inventory in process and expected lead time; and iv) discuss the application. Computer simulation is the research method. Delimitation is that made in a single application in shoe manufacturing, in a period of two weeks. The working method includes two operations research techniques, Petri nets (PN) and the throughput diagram and was tested in a production plan already performed, whose results served to refine and validate the model,

The main contribution of this paper is the method of working, replicable to other applications: simulation PN, validated by data field and use the throughput diagram results to calculate the performance metric. The method can be useful in ill-structured problems, as

The PN describes the system structure as a directed graph and can capture precedence relations and structural links of real systems with graphical expressiveness to model conflicts and queues. Formally, it can be defined as a sixfold (P, T, A, M0, W, K) in wich: P is a set of states/places, T is a set of transitions, A is a set of arcs subject to the constraint that arcs do not connect directly two positions or transitions, M0 is the initial state, which tells how many marks/tokens there are in each position to the beginning of the processing, W is a set of arc weights, which tells, for each arc, how many marks are required for a place by the transition or how many are placed in a place after the respective transition; and K is a set of capacity constraints, which reports to each position, the maximum number of marks which may occupy the place (Castrucci; Moraes, 2001). Applying the definition in the PN of Figure 1, P = [p0, p1]; T = [t0]; A = [(p0, t0), (t0, p1)]; W: w (p0, t0) = 1, w (t0, p1) = 1 e M0 = [1; 0]. The

The transitions correspond to changes of states and places correspond to state variables of the system. In the firing of a transition, the tokens move across the network in two phases: enabling and firing transition. A transition tj ∈ T is enabled by a token m if ∀ pi ∈ P, m (pi) ≥ w(pi, tj), i.e., the token in place pi is greater than or equal to the arc weight that connects pi to tj. Some variations are allowed in Petri Nets and were used for modeling, for example, the use

manufacturing and the results compared with the real manufacturing outputs.
