Combining Logic and Optimization in Cutting Plane Theory

AbstractWe consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in K that are not lexicographically smaller than x. The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program $$\min \{cx: x\in S\cap \mathbb {Z}^n\}$$ min { c x : x ∈ S ∩ Z n } , where $$S\subset \mathbb {R}^n$$ S ⊂ R n is a compact set and $$c\in \mathbb {Z}^n$$ c ∈ Z n . We analyze the number of iterations of our algorithm.

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Significance of cutting plane in liquid metal embrittlement severity quantification

SN Applied Sciences ◽

10.1007/s42452-021-04608-2 ◽

2021 ◽

Vol 3 (6) ◽

Author(s):

C. DiGiovanni ◽

L. He ◽

C. Hawkins ◽

N. Y. Zhou ◽

E. Biro

Keyword(s):

Liquid Metal ◽

Cross Section ◽

Standard Procedure ◽

Fuel Efficiency ◽

Zinc Coating ◽

High Strength Steels ◽

Absolute Measurement ◽

Cutting Plane ◽

Liquid Metal Embrittlement ◽

Welding Parameters

AbstractThe automotive industry is turning to advanced high strength steels (AHSS) to reduce vehicle weight and increase fuel efficiency. However, the zinc coating on AHSS can cause liquid metal embrittlement (LME) cracking during resistance spot welding. To understand the problem, the severity of the cracking must be measured. Typically, this is done from the weld cross-section. Currently, there is no standard procedure to determine which plane through the weld must be examined to gauge cracking severity, leading to a variety of practices for choosing a cutting plane. This work compares the magnitude and variability of LME severity measured from the plane of exhibiting the most severe surface cracking to arbitrarily chosen planes. The plane exhibiting the most severe cracks had more and longer cracks on the cross-section than the arbitrarily chosen plane, resulting in a higher crack severity measurement. This higher absolute measurement increased the relative accuracy of the examination, allowing for fewer welds to be examined to precisely determine the effect of LME mitigation methods on cracking severity, how welding parameters affect LME cracking severity and the predicted LME affected strength of a particular weld.

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