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S-graph Framework in Batch Process Scheduling T. Holczinger 1, R. Adonyi 1, G. Biros 1, J. Romero 2, L. Puigjaner 2, F. Friedler 1 1 Department of Computer Science, University of Veszprém, Veszprém, Egyetem u. 10., H-8200, Hungary 2 Department of Chemical Engineering, Universitat Politècnica de Catalunya, Barcelona, Av. Diagonal 647, E-08028 CAPE Forum 2004 Veszprem

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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 data Aim: The optimal order of the tasks Using the given resources

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Conventional representation Notations PT: processing time Eq.: equipment unit

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Graph representation

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Recipe Directed graphProducts: A, B, and C

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S-graph representation Directed graph The changeovers are denoted by arcs 2

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NIS vs. UIS 12 6 E1 E2 34 7 E1 12 6 E2 34 7 E1 Unlimited Intermediate Storage Non Intermediate Storage

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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) 123 456 7 8 9 E1E2E3 E1E2E3 E1E2E3 A B C 10 11 12

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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

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More than one batch per product Considering each batch as an individual product is not efficient enough Product A Product B

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Single equipment unit for a task

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Optional equipment units for a task with identical processing time

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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

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Illustrative example 12 7 34 8 5 6 9 E1E2E2 E2 E1E1E2E2 1 12 7 34 8 5 6 9 E1E2E2 E2 E1E1E2E2 1 2 3 1 2 3 12 7 34 8 5 6 9 E1E2E2 E2 E1E1E2E2 12 7 34 8 5 6 9 E1E2E2 E2 E1E1E2E2 1 2 3 4 4 S-graphSearch treeS-graphSearch tree

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Illustrative example 1 2 34 5 12 7 34 8 5 6 9 E1E2E2 E2 E1E1E2E2 12 7 34 8 5 6 9 E1E2E2 E2 E1E1E2E2 1 2 34 5 6 6 S-graphSearch tree

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Illustrative example: search tree With cycle prediction Without cycle prediction 1 2 34 5 6 7 12 7 34 8 5 6 9 E1E2E2 E2 E1E1E2E2 1 2 34 5 6 7 1 2 34 5 6 7 8 9 10 111213

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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)

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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%

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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/http://www.dcs.vein.hu/CAPO/demo/sch/

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