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A Lookahead Heuristic for the Minimization of Open Stacks Problem

A Lookahead Heuristic for the Minimization of Open Stacks Problem. Marco A. M. Carvalho mamc@ iceb.ufop.br Federal University of Ouro Preto - Brazil Nei Y. Soma soma@ ita.br Technological Institute of Aeronautics - Brazil OR54 Annual Conference – Edinburgh, UK 04-06 September 2012.

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A Lookahead Heuristic for the Minimization of Open Stacks Problem

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  1. A Lookahead Heuristic for the Minimization of Open Stacks Problem Marco A. M. Carvalho mamc@iceb.ufop.br Federal University of OuroPreto - Brazil Nei Y. Soma soma@ita.br Technological Institute of Aeronautics - Brazil OR54 Annual Conference – Edinburgh, UK 04-06 September 2012

  2. Problem Description Motivation Example Introduction OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  3. Introduction • A factorymanufacturesdifferenttypesofproducts in batches; • Customersplaceorders for differentproducts • The contentsofeachorder are placed in a separatedstackduringmanufacturing; • Whenthestackreceivesthefirstproduct, it isopened; • Whenthestackreceivesthelastproduct , it isclosed • The products are delivered; • The spaceisfreed. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  4. Introduction • Thereis a limitationonthephysicalspaceused in theproductionenvironment • Thereisnotenoughspace for allcustomer’sstackstobeopenedsimutaneously; • Ifthenumberofopen stacksincreasesbeyondtheavailablespace, stacks must beremoved in ordertogivespacetothe new stacks. • The sequence in whichtheproducts are manufacturedcanreducethemaximumnumberofsimultaneously open stacks • ThisistheaimoftheMinimizationof Open StacksProblem (MOSP). OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  5. Motivation • The problemis NP-Hard andhas a varietyofequivalentproblems: • Cutting stock • CuttingPatternsSequencing. • VLSI design • Gate Matrix Layout Problem; • PLA Folding. • GraphProblems • Pathwidth; • IntervalThickness; • Node Search Game; • Narrowness; • Split bandwidth; • Edge andVertexSeparation. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  6. Example #1 Sixcustomers; Sixproducttypes. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  7. Example #1 OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK Manufacturing Sequence: Open Stacks: 0

  8. Example #1 OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK Manufacturing Sequence: Open Stacks: 3

  9. Example #1 OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK Manufacturing Sequence: Open Stacks: 4

  10. Example #1 OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK Manufacturing Sequence: Open Stacks: 5

  11. Example #1 OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK Manufacturing Sequence: Open Stacks: 4

  12. Example #1 OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK Manufacturing Sequence: Open Stacks: 4

  13. Example #1 OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK Manufacturing Sequence: Open Stacks: 2

  14. Introduction • Formally, given a boolean matrix M: • Rows correspond to customer’s orders; • Columns correspond to products; • mij = 1 iff order i contains product j; • mij = 0 otherwise; • Stacks are associated to rows • First product is manufactured: stack opened; • Last product is manufactured: stack closed; • The objective is to find a permutation of columns such that the maximum number of open stacks is minimized. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  15. Example #1 Revisited OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  16. Example #1 Revisited Manufacturing Sequence Manufacturing Sequence Stacks Stacks Max Open Stacks: 4 Max Open Stacks: 6 OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  17. Representation Preprocessing Breadth-First Search Products Sequencing Improvement Rules A Lookahead heuristic OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  18. Representation • In MOSP graphs, nodes correspond to customer’s orders • Edges connect customers that ordered at least one product in common; • Multiple edges and loops are not considered; • Each product produces a clique in the graph; • There are polynomial algorithms for some special topologies. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  19. MOSP Graph OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  20. MOSP Graph OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  21. MOSP Graph OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  22. MOSP Graph OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  23. MOSP Graph OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  24. MOSP Graph OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  25. MOSP Graph OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  26. Preprocessing #1 Ifthe MOSP graphisdisconnected, theproblemisdecomposable. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  27. Preprocessing #2 • Letc(pi) determine the set ofcustomersthatorderedproductpi • Ifc(pj) ⊆c(pi), thenpiandpjcanbeconsidered as oneproduct. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  28. Breadth-First Search • The MOSP resembles the Matrix Bandwidth Minimization Problem (MBM) • The MBM problem aims to find a permutation of rows and columns which keeps the nonzero elements of a matrix as close as possible to the main diagonal. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  29. Breadth-First Search • The Cuthill-Mckee (1969) heuristic for MBM explores a corresponding graph by Breadth-First Search (BFS): • Choice of lower degree nodes • Ties are broken in favor of the lower index node; • The sequence of the search determines the permutation of rows and columns. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  30. Breadth-First Search • The BFS has never been applied to the MOSP solution • MOSP instances may not be sparse, symmetric or square, as MBM matrices; • The band structure is not a required condition. • However, when applied to the MOSP, it generates good results. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  31. Breadth-First Search Notexamined Examined Allneighborsexamined OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  32. Products Sequencing • After sequencing the nodes (orders), we obtain the products permutation: • The orders are analyzed using LIFO policy; • Each ordered product is inserted in the solution using LIFO policy. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  33. Products Sequencing Q={3, 1, 4, 5, 2, 6} OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  34. Products Sequencing Q={3, 1, 4, 5, 2, 6} OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  35. Products Sequencing Q={3, 1, 4, 5, 2, 6} OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  36. Products Sequencing Q={3, 1, 4, 5, 2, 6} OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  37. Products Sequencing Manufacturing Sequence Stacks Max Open Stacks: 4 OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  38. Breadth-First Search • Breadth-First Search features: • Low degree nodes are not the problem’s bottleneck • Sequenced first. • Clique’s and high degree nodes tend to be sequenced contiguously; • Preprocessing #1 is inherent; • Computational complexity; • Ease of implementation. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  39. Improvement Rules • Special topologies of the MOSP graph cause BFS to generate errors: • Cliques loosely connected; • A dominant clique with a few nodes in its neighborhood. • Improvement rules: • Close inactive open stacks, by anticipating its product's manufacturing; • Delay the opening of new stacks, by postponing its product’s manufacturing. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  40. ImprovementRule #1 Manufacturing Sequence Manufacturing Sequence Stacks Stacks Max Open Stacks: 6 Max Open Stacks: 5 OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  41. ImprovementRule #2 Manufacturing Sequence Manufacturing Sequence Stacks Stacks Max Open Stacks: 6 Max Open Stacks: 5 OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  42. Data sets Computational Environment Computational experiments OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  43. Datasets • First Constraint Modeling Challenge (2005) • 5,806 smaller instances; • Decomposable instances; • Polynomial topologies of MOSP graphs. • Harder Instances (2009) • 200 larger instances; • No decomposable instances; • No polynomial topologies of MOSP graphs. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  44. ComputationalExperiments • Intel i5 Quad Core 3.2 GHz processor; • 16 GB RAM; • Ubuntu 12.4.1; • No optimizationoptions; • Chu andStuckey (2009) original code, compiledandrun as recommended • MOSP state-of-the-artexactmethod. • ImplementationofBecceneri, Yanasseand Soma (2004) as originallydescribed • MOSP state-of-the-artheuristic. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  45. Dataset #1 • Solutions Running times (ms) OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  46. Dataset #1 OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  47. Dataset#2 • Solutions Running times (ms) OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  48. Dataset#2 OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  49. summary OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

  50. Summary • A novel approach to MOSP; • O(p3) heuristic, where p denotes the number of products • Outperforms the state-of-the art heuristic in solution quality • Smaller gaps from optimal; • Robust - smaller errors; • Higher index of optimal solutions. • Fast. • Can be used to generate good upper bounds; • Can be used directly to solve MOSP and equivalent problems. OR54 Annual Conference, 04-06 September 2012 – Edinburgh, UK

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