Exact and heuristic solutions for combinatorial optimization problems

Díaz, José Carlos Díaz

doi:10.1007/s10288-013-0239-y

Cited by 2 publications

(7 citation statements)

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“…By enhancing the enumeration approach to a B&B algorithm we were able to computationally prove that all 1CSP instances with demand of at most 9 are proper IRUP instances, while we found classes of non-IRUP instances with demand n = 10 and ∆ p = 1. This resolves an open question from [2,4], where the authors ask for proper non-IRUP instances with n < 13. ∆ p > 1 is possible for n ≥ 11 only.…”

Section: Resultsmentioning

confidence: 91%

“…The example of [16] has L = 3, 397, 386, 355 and was decreased to just L = 1, 111, 139 in [3]. Recently the authors of [2,4] gave an example with L = 100. Our smallest found example has L = 18.…”

Section: Resultsmentioning

confidence: 99%

“…Due to ∆ p (E) ≤ ∆(E) proper non-IRUP instances are also non-IRUP instances. The converse is not true as shown by E = 3, 30, (2,3,5) , (1,2,4) with ∆(E) = 31/30 and ∆ p (E) = 4/5.…”

Section: Basic Notationmentioning

confidence: 97%

“…If the proper gap is at least 1, we speak of a proper non-IRUP instance. The currently largest known proper gap of 1CSP is given by 1.003 [2,4].…”

Section: Introductionmentioning

confidence: 99%

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Minimal proper non-IRUP instances of the one-dimensional cutting stock problem

Kartak

Ripatti

Scheithauer

et al. 2015

Discrete Applied Mathematics

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We consider the well-known one dimensional cutting stock problem (1CSP). Based on the pattern structure of the classical ILP formulation of Gilmore and Gomory, we can decompose the infinite set of 1CSP instances, with a fixed demand n, into a finite number of equivalence classes. We show up a strong relation to weighted simple games. Studying the integer round-up property we computationally show that all 1CSP instances with n ≤ 9 are proper IRUP, while we give examples of a proper non-IRUP instances with n = 10. A gap larger than 1 occurs for n = 11. The worst known gap is raised from 1.003 to 1.0625. The used algorithmic approaches are based on exhaustive enumeration and integer linear programming. Additionally we give some theoretical bounds showing that all 1CSP instances with some specific parameters have the proper IRUP.Keywords: bin packing problem, cutting stock problem, integer round-up property, equivalence of instances, branch and bound method, linear programming, weighted simple games.

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Section: Resultsmentioning

confidence: 91%

Section: Resultsmentioning

confidence: 99%

Section: Basic Notationmentioning

confidence: 97%

“…If the proper gap is at least 1, we speak of a proper non-IRUP instance. The currently largest known proper gap of 1CSP is given by 1.003 [2,4].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Minimal proper non-IRUP instances of the one-dimensional cutting stock problem

Kartak

Ripatti

Scheithauer

et al. 2015

Discrete Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…For a demand of at most 9 the corresponding numbers are determined. As a side result we could correct the number of weighted simple games for 9 4 Each 0 < ε ≤ [11]. To the best of our knowledge, the relation between 1CSP instances and weighted simple games is indicated for the first time.…”

Section: Resultsmentioning

confidence: 99%