**3. Throughput diagram in manufacturing**

In manufacturing, a queue arises when, for variability, at a given instant, the number of orders to be implemented is greater than the available job centers. The manufacturing arrives at the work position (or center), waiting its turn, is processed and proceeds. The sequence is subject to change priorities and interruptions for maintenance or lack of materials (Silva; Morabito, 2007; Papadopoulos et al., 1993).

A work center (machine, production line or manufacturing plant) can be compared to a funnel, in which orders arrive (input), waiting for service (inventory) and leave the system (output). When the work center is observed for a continuous period, the reference period, the cumulative results can be plotted. In Figure 2, it is possible to observe strokes representing the accumulated input and output, measured in amount of work (Wiendahl, 1995). This quantity may be in parts, numbers of hours or another unit value which represents a significant manufacturing effort (Sellitto, 2005).

**Figure 2.** Throughput diagram of a work center

To obtain the line that represents input is necessary knowing the amount of work waiting in the initial inventory at the beginning of the reference period and the output is plotted summing the completed work orders. Wiendahl (1995) presents an analytical development related to the throughput diagram and calculates various quantities of interest to workload control (WLC) such as: lead time, average performance, autonomy, work progress and delays in delivery of orders. For this study case, the funnel formula will be applied:

$$\text{TLm} = \text{I}\_{\text{m}} / \text{P}\_{\text{m}} \tag{1}$$

Measurement of Work-in-Process and Manufacturing Lead Time by Petri Nets Modeling and Throughput Diagram 99

To assign time to the transitions, all the processes were timed. With the orientation of the production supervisor, the start/end time of each task was defined. It was considered a confidence level of 95% and used the calculation model suggested by Vaz (1993) and AEP (2003). As an example, for the sole cutting operation the average time was 18.13 seconds and the standard deviation was 2 seconds. The minimum number of samples to ensure the confidence level was 19.5, adopting 20 samples. The time for each transition is the average

To test and refine the model was chosen a plan already done, two weeks and nine production orders. It was informed the load for each place, resulting from earlier orders, at the moment of the first evaluated order will enter the system. A new order is queued of previous processing orders, which explains why the lead time in manufacturing is much higher than the standard manufacturing time. The queuing discipline adopted was FIFO (First-In-First-Out). Table 1 shows data from nine manufacturing orders contained in the production plan (dates are considering working days – 8h40m/day = 3.200s/day). In the last

column, there is the order lead time, calculated by simulation, and their average.

Real Input Date (s)

**Table 1.** Information for inventory and mean lead time calculation in manufacturing

comparison of information from real and simulated outputs, order to order.

Wiendahl (1995) presents a method that considers the size of the order Qi. By this method, TLm = [ ΣQi x TLorder I ] / ΣQi = 2.73 days, close to the calculated 2.63 days. The correlation between real and simulated outputs (column 5 and 6) is 0.99 and the absolute error | real simulated | average is 9,821s (2.27% of the largest real value). Figure 3 shows the

1,000 0 2.5 0 78,000 66,480 66,480 500 1 3.5 31,200 109,200 104,450 73,250 1,500 2 6.5 62,400 202,800 174,860 112,460 800 4 7.5 124,800 234,000 233,872 109,072 800 5 9 156,000 280,800 280,704 124,704 400 7 10 218,400 312,000 313,763 95,363 1,000 10.2 12.5 318,240 390,000 361,404 43,164 500 11.2 13 349,440 405,600 399,304 49,864 500 11.7 14 365,040 436,800 428,504 63,464 TOTAL 7,000 Pairs AVERAGE 81,980 s = 2,627 days

Real Output Date (s) Simulated Output Date (s)

Simulated Lead Time of Order (s)

**4.2. Transition time assignments** 

**4.3. Simulation, inventory and lead time calculations** 

Real Output Date (days)

values collected.

Order (pairs)

Real Input Date (days)

Where, TLm = simple mean lead time of orders (days); Im = mean inventory (parts); and Pm = mean performance (parts per day).

For the demonstration, the author uses the figure and considers steady state, i.e., the balance between input and outputs (α1 = α2) and tan α1 = Im / TLm e tan α2 = Pm. Wiendahl (1995) suggests that the equation can be used for measurement and control of manufacturing.
