Online packing of d-dimensional boxes into the unit cube

Packing of points into the unit 6-dimensional cube

The best known estimate for the packing of points into the 6-dimensional unit cube in mutual distances at least 1 is improved. The following problem was stated in [10] and later repeated e.g. in [1], [6], [11], [12], [13]. “Let f(n) denote the maximum number of points that can be arranged in the n-dimensional unit cube so that all mutual distances are at least 1. Obviously, f(n) = 2n for n ≤ 3....

Large Triangles in the d-Dimensional Unit-Cube

We consider a variant of Heilbronn’s triangle problem by asking for fixed dimension d ≥ 2 for a distribution of n points in the d-dimensional unit-cube [0, 1] such that the minimum (2-dimensional) area of a triangle among these n points is maximal. Denoting this maximum value by Δoff−line d (n) and Δ on−line d (n) for the off-line and the online situation, respectively, we will show that c1 ·(l...

Online strip packing with modifiable boxes

Fast Truck - Packing of 3 D boxes

The purpose of this research is to estimate the repair effects a repairable system undergoes repeated failures and repairs. Data was collected on a cement kiln incurring150 failures and 5 shutdowns for overhaul. Causes of failure were identified as operational, mechanical or electrical. The unit was restarted but with an (expected) improved failure rate by virtue of the repair. Repair improveme...

Online Multi-dimensional Dynamic Bin Packing of Unit-Fraction Items

We study the 2-D and 3-D dynamic bin packing problem, in which items arrive and depart at arbitrary times. The 1-D problem was first studied by Coffman, Garey, and Johnson motivated by the dynamic storage problem. Bar-Noy et al. have studied packing of unit fraction items (i.e., items with length 1/k for some integer k ≥ 1), motivated by the window scheduling problem. In this paper, we extend t...