S-graph Framework in Batch Process Scheduling

T. Holczinger1, R. Adonyi1, G. Biros1,

J. Romero2, L. Puigjaner2, F. Friedler1

1Department of Computer Science, University of Veszprém, Veszprém, Egyetem u. 10., H-8200, Hungary

2Department of Chemical Engineering, Universitat Politècnica de Catalunya, Barcelona, Av. Diagonal 647, E-08028

CAPE Forum 2004

Veszprem

Problem definition

Given:• The order of the tasks (recipe)• The set of plausible equipment units for each task

(with operation times)• Necessary amount of products• Storage policy• Timing dataAim:• The optimal order of the tasks• Using the given resources

Conventional representation

Notations

PT: processing time

Eq.: equipment unit

Graph representation

Product A

Product B

Product C

Recipe

• Directed graph Products: A, B, and C

S-graph representation

• Directed graph• The changeovers are denoted by arcs

2

NIS vs. UIS

1 26E1 E2

3 47E2 E1

1 26E1 E2

3 47E2 E1

Unlimited Intermediate Storage

Non Intermediate Storage

Schedule-graph

• An S-graph is a schedule-graph, if– the equipment-task assignment and

– the operation order of the tasks are given (schedule-arc)

1 2 3

4 5 6

7 8 9

E1 E2E3

E1 E2 E3

E1E2 E3

A

B

C

10

11

12

Basic algorithm

• Branch and bound framework• Extension of the recipe-graph with schedule-arcs

according to the rules– Introduction of schedule-arcs

– Identical set of nodes

– The weight of the recipe-arcs can be changed

– The number of the possible extensions are finite the algorithm is finite

More than one batch per product

• Considering each batch as an individual product is not efficient enough

Product A

Product B

Single equipment unit for a task

Optional equipment units for a task with identical processing time

Acceleration tools

• Cycle prediction– To predict that a subgraph has no feasible solution

– Based on the cycle search algorithm

– It can reduce the size of the search tree

• LP model for sharpening the lower bound

Illustrative example

1 2 7

3 4 8

5 6 9

E1 E2

E1 E2

E1 E2

11 2 7

3 4 8

5 6 9

E1 E2

E1 E2

E1 E2

1

2

3

1

2 3

1 2 7

3 4 8

5 6 9

E1 E2

E1 E2

E1 E2

1 2 7

3 4 8

5 6 9

E1 E2

E1 E2

E1 E2

1

2 3 44

S-graph Search tree S-graph Search tree

Illustrative example

1

2 3 4

5

1 2 7

3 4 8

5 6 9

E1 E2

E1 E2

E1 E2

1 2 7

3 4 8

5 6 9

E1 E2

E1 E2

E1 E2

1

2 3 4

5 66

S-graph Search tree

Illustrative example: search tree

With cycle prediction Without cycle prediction

1

2 3 4

5 6

7

1 2 7

3 4 8

5 6 9

E1 E2

E1 E2

E1 E2

1

2 3 4

5 6

7

1

2 3 4

5 6 7 8 9 10

11 12 13

Industrial size scheduling problem

• 123 products– 31 different base products– 13 possible pack sizes from 5 to 20000 liters– Necessary amount is from 5 to 12000 tons for a year (solved the

production of a week)

• Batch size is 6 tons for each mixer• Two types of tank

– T901 – T922 (8 tons)– T951 – T968 (15 tons)

• 120 minutes minimum residence time for intermediate products in a tank (bubbling)

Parameters of the problem

Number of products: 123

Total number of batches: 389

Number of equipment units

Mixer: 5 (batch type equipment unit)

Tank: 40

Packing line: 26 (continuous type equipment unit)

Running time on PC (1 GHz) is less than 4 minutes and the optimality gap is 2.6%

Further information

• Publications– E. Sanmartí, F. Friedler, and L. Puigjaner, Combinatorial technique for short

term scheduling of multipurpose batch plants based on schedule-graph representation, Comput. Chem. Engng. 22 (1998)

– E. Sanmartí, L. Puigjaner, T. Holczinger, and F. Friedler, Combinatorial framework for effective scheduling of multipurpose batch plants, AIChE J. 48 (2002)

– Holczinger, T., J. Romero, L. Puigjaner, and F. Friedler, Scheduling of Multipurpose Batch Processes with Multiple Batches of the Products, Hung. J. Ind. Chem., 30, 305-312 (2002)

• Demonstration programs– http://www.dcs.vein.hu/CAPO/demo/sch/