Packing the maximum number of m × n tiles in a large p × q rectangle

Barnes, Frank W.

doi:10.1016/0012-365x(79)90115-8

Cited by 34 publications

(10 citation statements)

References 2 publications

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“…We call this bound the Minimum Area Ratio (MAR) Bound. Barnes (1979) proposes a bound based on boxes being represented by patterns of a · 1 or b · 1 boxes. Because the solution to problems with unit width is easily obtained (Barnett and Kynch, 1967), Barnes computes the wasted area obtained when placing only a · 1 or only b · 1 boxes to produce an upper bound on the number of boxes placed.…”

Section: Easily Computed Boundsmentioning

confidence: 99%

Solving the pallet loading problem

Martins

Dell

2008

European Journal of Operational Research

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Section: Easily Computed Boundsmentioning

confidence: 99%

Solving the pallet loading problem

Martins

Dell

2008

European Journal of Operational Research

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“…Other lower and upper bounds better than (8) or (9) might be de®ned; see the discussion in 20 . In particular, an interesting upper bound was proposed in Barnes 21 which is based on the consideration that each packing of an lY w rectangle on LY W is also a packing of either lY 1 or 1Y w tiles on LY W . A formal algorithm for the B&D heuristic is presented belowÐthe solution produced by this algorithm with parameters LY W and lY w, or W Y L and lY w, corresponds to the best 1st-order non-guillotine pattern of, at most, ®ve blocks.…”

Section: Approximate Methods and The Bandd Heuristicmentioning

confidence: 99%

A simple and effective recursive procedure for the manufacturer's pallet loading problem

Morábito

Morales

1998

Journal of the Operational Research Society

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In this paper we present a simple and effective heuristic to solve the problem of packing the maximum number of rectangles of sizes lY w and wY l into a larger rectangle LY W without overlapping. This problem appears in the loading of identical boxes on pallets, namely the manufacturer's pallet loading (MPL), as well as in package design and truck or rail car loading. Although apparently easy to be optimally solved, the MPL is claimed to be NP-complete and several authors have proposed approximate methods to deal with it. The procedure described in the present paper can be seen as a re®nement of Bischoff and Dowsland's heuristic and can easily be implemented on a microcomputer. Using moderate computational resources, the procedure was able to ®nd the optimal solution of 99.9% of more than 20 000 examples analysed.

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“…The first two improvements consist of the usage of raster points and the upper bound introduced by Barnes (1979). The raster points are incorporated in connection with the data structures described in Birgin et al (2005).…”

Section: Refinements Of the Recursive Five-block Heuristicmentioning

confidence: 99%

“…Various authors have proposed approximate methods to deal with it, as discussed in, for example, Balasubramanian (1992), Herbert and Dowsland (1996), Scheithauer and Terno (1996), Scheithauer and Sommerweiss (1998), Letchford and Amaral (2001), Maing-Kyu and Young-Gun (2001), Martins (2003), Alvarez-Valdes et al (2005a, b), Pureza and Morabito (2006) and the references therein. Upper bounds for the problem were studied in, for example, Barnes (1979); Dowsland (1985), Nelißen (1995), Letchford and Amaral (2001), Morabito and Farago (2002), Martins (2003).…”

Section: Introductionmentioning

confidence: 99%

An effective recursive partitioning approach for the packing of identical rectangles in a rectangle

Birgin

Lobato

Morábito

2010

Journal of the Operational Research Society

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In this work, we deal with the problem of packing (orthogonally and without overlapping) identical rectangles in a rectangle. This problem appears in different logistics settings, such as the loading of boxes on pallets, the arrangements of pallets in trucks and the stowing of cargo in ships. We present a recursive partitioning approach combining improved versions of a recursive five-block heuristic and an L-approach for packing rectangles into larger rectangles and L-shaped pieces. The combined approach is able to rapidly find the optimal solutions of all instances of the pallet loading problem sets Cover I and II (more than 50 000 instances). It is also effective for solving the instances of problem set Cover III (almost 100 000 instances) and practical examples of a woodpulp stowage problem, if compared to other methods from the literature. Some theoretical results are also discussed and, based on them, efficient computer implementations are introduced. The computer implementation and the data sets are available for benchmarking purposes.

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Packing the maximum number of m × n tiles in a large p × q rectangle

Cited by 34 publications

References 2 publications

Solving the pallet loading problem

Solving the pallet loading problem

A simple and effective recursive procedure for the manufacturer's pallet loading problem

An effective recursive partitioning approach for the packing of identical rectangles in a rectangle

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