Journal of the Operational Research Society

, Volume 66, Issue 8, pp 1297–1311 | Cite as

A heuristic for the Minimum Score Separation Problem, a combinatorial problem associated with the cutting stock problem

  • Kai Helge Becker
  • Gautam Appa
General Paper

Abstract

The Minimum Score Separation Problem (MSSP) is a combinatorial problem that was introduced in JORS 55 as an open problem in the paper industry arising in conjunction with the cutting stock problem. During the process of producing boxes, flat papers are prepared for folding by being scored with knives. The problem is to determine whether and how a given production pattern of boxes can be arranged such that a certain minimum distance between the knives can be kept. Introducing the concept of matching-based alternating Hamiltonian paths, this paper models the MSSP as the problem of finding an alternating Hamiltonian path on a graph that is the union of a matching and a type of graph known as a ‘threshold graph’. On this basis, we find a heuristic that can quickly recognize a large percentage of feasible and infeasible instances of the MSSP. Detailed computational experiments demonstrate the efficiency of our approach.

Keywords

cutting stock problem bin packing heuristics networks and graphs Travelling Salesman Problem alternating Hamiltonian path threshold graph 

References

  1. Bang-Jensen J and Gutin G (1997). Alternating cycles and paths in edge-coloured multigraphs: A survey. Discrete Mathematics 165–166: 39–60.CrossRefGoogle Scholar
  2. Brandstädt A, Le VB and Spinrad JB (1999). Graph Classes—A Survey. SIAM monographs on discrete mathematics and applications SIAM: Philadelphia.CrossRefGoogle Scholar
  3. Chvátal V and Hammer PL (1973). Set Packing Problems and Threshold Graphs. CORR 73-21. University of Waterloo: Canada.Google Scholar
  4. Chvátal V and Hammer PL (1977). Aggregation of inequalities in integer programming. In: Hammer PL, Johnson EL, Korte BH and Nemhauser GL (eds). Studies in Integer Programming. Annals of Discrete Mathematics 1. North-Holland Publishing Company: Amsterdam, pp 145–162.CrossRefGoogle Scholar
  5. Cook WJ, Cunningham WH, Pulleyblank WR and Schrijver A (1998). Combinatorial Optimization. Wiley: New York.Google Scholar
  6. Ecker K and Zaks S (1977). On a Graph Labelling Problem. Bericht 99. Gesellschaft für Mathematik und Datenverarbeitung mbH: Bonn.Google Scholar
  7. Gilmore PC and Gomory RE (1961). A linear programming approach to the cutting-stock problem. Operations Research 9 (6): 849–859.CrossRefGoogle Scholar
  8. Gilmore PC and Gomory RE (1963). A linear programming approach to the cutting stock problem—Part II. Operations Research 11 (6): 863–888.CrossRefGoogle Scholar
  9. Golumbic MC (1980). Algorithmic Graph Theory and Perfect Graphs. Academic Press: New York.Google Scholar
  10. Goulimis C (2004). Minimum score separation—an open combinatorial problem associated with the cutting stock problem. Journal of the Operational Research Society 55 (12): 1367–1368.CrossRefGoogle Scholar
  11. Henderson PB and Zalcstein Y (1977). A graph-theoretic characterization of the PVchunk class of synchronizing primitives. SIAM Journal of Computing 6 (1): 88–108.CrossRefGoogle Scholar
  12. Karp RM (1972). Reducibility among combinatorial problems. In: Miller RE and Thatcher JW (eds). Complexity of Computer Computations. Plenum Press: New York, pp 85–103.CrossRefGoogle Scholar
  13. Koren M (1973). Extreme degree sequences of simple graphs. Journal of Combinatorial Theory B 15 (3): 213–224.CrossRefGoogle Scholar
  14. Laporte G, Asef-Vaziri A and Sriskandarajah C (1996). Some applications of the generalized Travelling Salesman Problem. Journal of the Operational Research Society 47 (12): 1461–1467.CrossRefGoogle Scholar
  15. Lewis R, Song X, Dowsland K and Thompson J (2011). An investigation into two bin packing problems with ordering and orientation implications. European Journal of Operational Research 213 (1): 52–65.CrossRefGoogle Scholar
  16. Mahadev NVR and Peled UN (1994). Longest cycles in threshold graphs. Discrete Mathematics 135 (1-3): 169–176.CrossRefGoogle Scholar
  17. Mahadev NVR and Peled UN (1995). Threshold Graphs and Related Topics. Annals of Discrete Mathematics 56. Elsevier: Amsterdam.Google Scholar

Copyright information

© Operational Research Society Ltd. 2014

Authors and Affiliations

  • Kai Helge Becker
    • 1
  • Gautam Appa
    • 2
  1. 1.Queensland University of Technology (QUT)BrisbaneAustralia
  2. 2.London School of Economics and Political ScienceLondonUK

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