**5.4. Modelling**

18 Will-be-set-by-IN-TECH

Product Code Amount Starting time Blue PB 6 0 Green PG 6 0

**5.3. Structure of the production facility given in production management system** Data about the production process and its structure can be obtained from the production management information systems. These data can be presented in a form of a recipe and

There are two recipes available, which are specifying a sequence of operations needed to

Op20 draining R2x in B31 12/12/12 [R\_1](R21/R22/R23), B31

Op20 draining R2x in B32 13/12/12 [R\_ 1](R21/R22/R23), B32

Op20 draining B11 in R2x 15/12/12 [R\_3](R21/R22/R23), [BY\_2](B11)

Op20 draining B13 in R2x 10/9/13 [R\_2](R21/R22/R23), [BW\_2](B13)

Op20 draining B12 in R2x 11/13/14 [R\_ 3](R21/R22/R23), [BR\_ 2](B12)

Op20 draining B13 v R2x 10/9/13 [R\_ 2](R21/R22/R23), [BW\_ 2](B13)

To produce Blue product ("PB"), firstly operation *Op10* has to be carried out. This operation determines only that a sub-product, defined with formula "B", is needed. When this sub-product is ready, two more operations are needed. *Op20* gives information about draining the product into the buffer tank *B31* and *Op30* about pumping the final product out of that buffer tank. The fact, that buffer tank could not be emptied before three batches of sub-product

In batch manufacturing situations when one task has to be executed with a (group of) resource(s), that were used to execute a previous task already are common. These resources are in our case labelled with an additional mark, i.e. code with serial number in square brackets is added in front of this resource(s). For example, this occurs in our case when one of the reactor is being used. Code [*R*\_1] is assigned to the reactor (from a group of reactors *R*2*x*) which is needed for operation *Op*20 when producing product "B" and indicates that this resource can now be released. Note, that this resource was assigned with an operation where

produce product "PB" and product "PG". They are given with tables 5 and 6. Operation Duration Resources PB Op10 – BOM\_B

> Op30 pumping from B31 30 (3 × 1)×B31 Y Op10 pumping Y in B11 12 [BY\_1](B11)

WB Op10 pumping W in B13 12 [BW\_1](B13)

Operation Duration Resources PG Op10 – BOM\_ G

Op30 pumping from B32 30 (3 × 1)×B32 R Op10 pumping Y in B12 12 [BR\_ 1](B12)

WG Op10 pumping W in B13 12 [BW\_ 1](B13)

"B" are poured in it, is described with this notation: (3 × 1) × *B*31.

code [*R*\_3] was used already (*Op10* of a sub-product "Y").

**Table 4.** Work orders.

the formula as described in chapter 3.4.

**Table 5.** Recipe of a product "PB".

**Table 6.** Recipe of a product "PG".

In this subchapter production process described previously is modelled with timed Petri nets. To build a model the algorithm from Chapter 3.5 is used. This model can later be used to schedule all the tasks, that are necessary to produce as much final products as required by work orders.

From work orders, given in table 4 it is recognised which and how much of each products are required. Let start with the procedure on building the Blue product ("PB"). When applying the first step of our algorithm, the PN structure shown in figure 12 is achieved.

**Figure 12.** PN model of PB product.

In the second step, additional information are added into this model (framed part of figure 12). Data about all the details are gathered from recipe list of product "PB" (table 5). Information about emptying the buffer tank *B*31 (*Op30*) and draining the reactor *R*2*x* (*Op20*) are added. As operation (*Op20*) needs resources, that are used by previous operations, not all details are added yet at this place. A model, shown in Figure 13 is achieved.

**Figure 13.** PN model of PB product.

#### 20 Will-be-set-by-IN-TECH 22 Petri Nets – Manufacturing and Computer Science Automated Petri-Net Modelling for Batch Production Scheduling <sup>21</sup>

Operation *Op10* (place *PPB*1*o p*) is defined with formula for item "B" (see table 7). It represents a mixing operation of two raw materials "Y" and "WB". Both sub-items are described with two operations, where precedence constraints are included as given with formula. Figure 14 shows how this formula information is modelled with Petri nets.

**Figure 14.** PN model of PB product with inserted BOM information.

In figure 15 some parts of a model given in figure 14 are simplified and information about the usage of rectors *R*2*<sup>x</sup>* is added.

pr2

p WB\_2\_2op

p Y\_2\_2op

9

tWB\_2\_2

WB, Op.1

**Figure 16.** PN model of PB product.

presented in table 8.

**Table 8.** Results.

tWB\_1

tPBin

p Y\_1op

p WB\_1op

Y, Op.1

tY\_1

12

tY\_ \_3 <sup>2</sup>

10

tWB\_2\_1

15

tY\_2\_1

12

tY\_2\_2

13

tWB\_2\_3

pr1

p WB\_2\_1op

p Y\_2\_1op

pr3

p WB\_2\_3op

all the production constraints and the duration of the whole process can be identified. A schedule of the batch process using SPT priority rule is given with Gantt chart (figure 18). The results were compared with the results obtained using various algorithms and are

> Algorithm Makespan SPT rule 315s LPT rule 331s Branch and Bound ([17]) 323s MS Project ([3]) 329s

p Y\_2\_3op

R22 B31

p PB2\_1op

p PB2\_2op

R21

Automated Petri-Net Modelling for Batch Production Scheduling 23

12

tPB2\_1

R23

12

p PB 2\_ \_3op

tPB2\_3

12

tPB2\_2

**Figure 15.** PN model of PB product.

As there are three possible reactors (*R*21, *R*<sup>22</sup> or *R*23) that can be used to perform these operations this model is extended with the structure given in figure 16.

With this procedure a detailed timed Petri-net model of the production of the blue product ("PB") is obtained. The same procedure was performed to model also the production of green product ("PG"), and a Petri net model given in figure 17 is achieved.

## **5.5. Results**

The resulting model was at the end verified using P-invariant analysis. We can find out eleven P-invariant, where eight of them refer to resources and three of them to the production routes.

In this way a Petri net model of a multiproduct batch plant was achieved on which different scheduling algorithms can be performed in order to obtain the most effective production. Petri-net simulation was used to evaluate different schedules of tasks that are needed to produce the desired amount of final products. Makespan was the performance measure of interest. The schedule allows an easy visualisation of the process and ensures that sufficient raw materials ("Yellow", "Red" and "White" batches) are available at the right time. It respects 22 Petri Nets – Manufacturing and Computer Science Automated Petri-Net Modelling for Batch Production Scheduling <sup>21</sup> Automated Petri-Net Modelling for Batch Production Scheduling 23

**Figure 16.** PN model of PB product.

all the production constraints and the duration of the whole process can be identified. A schedule of the batch process using SPT priority rule is given with Gantt chart (figure 18).

The results were compared with the results obtained using various algorithms and are presented in table 8.


**Table 8.** Results.

20 Will-be-set-by-IN-TECH

Operation *Op10* (place *PPB*1*o p*) is defined with formula for item "B" (see table 7). It represents a mixing operation of two raw materials "Y" and "WB". Both sub-items are described with two operations, where precedence constraints are included as given with formula. Figure 14

pr WB, Op.1

tPB1out tPBin

In figure 15 some parts of a model given in figure 14 are simplified and information about the

tY\_2

tWB\_2

pr

p Y\_2op

pWB\_2op

As there are three possible reactors (*R*21, *R*<sup>22</sup> or *R*23) that can be used to perform these

With this procedure a detailed timed Petri-net model of the production of the blue product ("PB") is obtained. The same procedure was performed to model also the production of green

The resulting model was at the end verified using P-invariant analysis. We can find out eleven P-invariant, where eight of them refer to resources and three of them to the production routes. In this way a Petri net model of a multiproduct batch plant was achieved on which different scheduling algorithms can be performed in order to obtain the most effective production. Petri-net simulation was used to evaluate different schedules of tasks that are needed to produce the desired amount of final products. Makespan was the performance measure of interest. The schedule allows an easy visualisation of the process and ensures that sufficient raw materials ("Yellow", "Red" and "White" batches) are available at the right time. It respects

WB, Op.2 =tPB2 pPB2op

WB, Op.2

tWB\_2

tY\_2

p Y\_2op

p WB\_2op

R2x B31

X

Y, Op.1 Y, Op.2

shows how this formula information is modelled with Petri nets.

p Y\_1op

p WB\_1op

tPB1out tPBin

operations this model is extended with the structure given in figure 16.

product ("PG"), and a Petri net model given in figure 17 is achieved.

tY\_1

tWB\_1

usage of rectors *R*2*<sup>x</sup>* is added.

WB, Op.1

tWB\_1

**Figure 15.** PN model of PB product.

**5.5. Results**

p Y\_1op

pWB\_1op

tY\_1

**Figure 14.** PN model of PB product with inserted BOM information.

Y, Op.1 Y, Op.2

**Figure 18.** Production schedule.

**Acknowledgements**

**Author details** Dejan Gradišar

Gašper Mušiˇc

*Jožef Stefan Institute, Slovenia*

*Faculty of Electrical Engineering, University of Ljubljana, Slovenia*

Fund.

A procedure for using existing data from production management systems to build the Petri-net model was developed. Timed Petri nets with the holding-duration principle of time implementation were used to model basic production activities. For the purposes of scheduling, different heuristic rules can be used within Petri net simulation. The applicability of the proposed approach was illustrated on a practical scheduling problem, where the data about the production facility is given with the formula and recipe. The model achieved with the proposed method was used to determine a schedule for production operations. The proposed method is an effective way to get an adequate model of the production process,

Automated Petri-Net Modelling for Batch Production Scheduling 25

which can be used to develop different analyses of the treated system, e.g. schedules.

The work was done in the frame of the Competence Centre for Advanced Control Technologies. Operation is partly financed by the Republic of Slovenia, Ministry of Education, Science, Culture and Sport and European Union (EU) - European Regional Development

**6. Conclusion**

**Figure 17.** PN model of the production plant.

**Figure 18.** Production schedule.
