Optimized Packings with Applications by Giorgio Fasano & János D. Pintér

Author:Giorgio Fasano & János D. Pintér
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

6.5.2.2 Sets of Potential Allocation Points

For the 2D allocation (cutting or packing) problem let , denote the optimal value function. This function is non-decreasing. Moreover, it is piecewise constant since only a finite number of different sequences of allocated pieces exists.

Clearly, if or , then the problem can be split into subproblems or the size of the raw material can be reduced.

Let S L ∗(γ, ℓ) and S W ∗(γ, w) denote the coordinates of the jump discontinuities in L- and W-direction of the optimal value function v. Our aim is to find supersets of S L ∗(γ, ℓ)j and S W ∗(γ, w) which are independent on γ i , i ∈ I.

Theorem 13.

For any , and , we have

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