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Irregular packing: MILP model based on a polygonal enclosure

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Abstract

This paper addresses the two-dimensional irregular packing problem, also known as the nesting problem. This is a subset of cutting and packing problems of renowned practical and theoretical relevance. A mixed integer-linear programming formulation is proposed to optimize the packing of particular polygonal shapes, convex forms with 3–8 sides, since their opposite sides are parallel. The model can be used to pack enclosures of general irregular shapes, generating upper bounds to the optimal solutions. The model was tested with 270 mass generated instances of small dimensions.

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Authors and Affiliations

  1. Escola Politécnica/Universidade de São Paulo, Av. Prof. Almeida Prado, Travessa 2, No 128, Cidade Universitária, São Paulo, SP, Brazil

    Miguel Cezar Santoro & Felipe Kesrouani Lemos

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  1. Miguel Cezar Santoro
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  2. Felipe Kesrouani Lemos
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Correspondence to Felipe Kesrouani Lemos.

Appendix: Details of instances

Appendix: Details of instances

The instances run in Sect. 4 (“computational results”) are described in detail. In Table 5 the characteristics of instances are shown. In Table 6, we present the data about each item.

It is important to stress that \(\hbox {k}_{\mathrm{a}}\) and \(\hbox {k}_{\mathrm{b}}\) are parameters of each instance, since the model requires parallel edges among all items. A consequence of this fact is that \(\hbox {m}_{\mathrm{pi}}\), \(\hbox {k}_{\mathrm{a}}\) and \(\hbox {k}_{\mathrm{b}}, \hbox {n}_{\mathrm{pi}}\) are given.

Table 5 Characteristics of the instances
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Table 6 Characteristics of the items
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Finally, Table 7 shows the results obtained for each instance of the mass generated set, for further comparison as benchmark.

Table 7 Results obtained for mass generated instances
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Santoro, M.C., Lemos, F.K. Irregular packing: MILP model based on a polygonal enclosure. Ann Oper Res 235, 693–707 (2015). https://doi.org/10.1007/s10479-015-1971-9

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