Minimizing setups and waste when printing labels of consumer goods

A real-world planning problem of a printing company is presented where different sorts of a consumer goods’ label are printed on a roll of paper with sufficient length. The printer utilizes a printing plate to always print several labels of same size and shape (but possibly different imprint) in parallel on adjacent lanes of the paper. It can be decided which sort is printed on which (lane of a) plate and how long the printer runs using a single plate. A sort can be assigned to several lanes of the same plate, but not to several plates. Designing a plate and installing it on the printer incurs fixed setup costs. If more labels are produced than actually needed, each surplus label is assumed to be “scrap”. Since demand for the different sorts may be heterogeneous and since the number of sorts is usually much higher than the number of lanes, the problem is to build “printing blocks”, i.e., to decide how many and which plates to design and how long to run the printer with a certain plate so that customer demand is satisfied with minimum costs for setups and scrap. This industrial application is modeled as an extension of a so-called job splitting problem which is solved exactly and by various decomposition heuristics, partly basing on dynamic programming. Numerical tests compare both approaches with further straightforward heuristics and demonstrate the benefits of decomposition and dynamic programming for large problem instances.

Entanglement entropy in cubic gravitational theories

We derive the holographic entanglement entropy functional for a generic gravitational theory whose action contains terms up to cubic order in the Riemann tensor, and in any dimension. This is the simplest case for which the so-called splitting problem manifests itself, and we explicitly show that the two common splittings present in the literature — minimal and non-minimal — produce different functionals. We apply our results to the particular examples of a boundary disk and a boundary strip in a state dual to 4- dimensional Poincaré AdS in Einsteinian Cubic Gravity, obtaining the bulk entanglement surface for both functionals and finding that causal wedge inclusion is respected for both splittings and a wide range of values of the cubic coupling.

R-symmetric flipped SU(5)

We construct a supersymmetric flipped SU(5) grand unified model that possesses an R symmetry. This R symmetry forbids dangerous non-renormalizable operators suppressed by a cut-off scale up to sufficiently large mass dimensions so that the SU(5)-breaking Higgs field develops a vacuum expectation value of the order of the unification scale along the F- and D-flat directions, with the help of the supersymmetry-breaking effect. The mass terms of the Higgs fields are also forbidden by the R symmetry, with which the doublet-triplet splitting problem is solved with the missing partner mechanism. The masses of right-handed neutrinos are generated by non-renormalizable operators, which then yield a light neutrino mass spectrum and mixing through the seesaw mechanism that are consistent with neutrino oscillation data. This model predicts one of the color-triplet Higgs multiplets to lie at an intermediate scale, and its mass is found to be constrained by proton decay experiments to be ≳ 5 × 1011 GeV. If it is ≲ 1012 GeV, future proton decay experiments at Hyper-Kamiokande can test our model in the p → π0μ+ and p → K0μ+ decay modes, in contrast to ordinary grand unified models where p → π0e+ or p → $$ {K}^{+}\overline{\nu} $$ K + ν ¯ is the dominant decay mode. This characteristic prediction for the proton decay branches enables us to distinguish our model from other scenarios.

Research of Data Processing Algorithms from Fiber Optic Sensors

The article considers the phenomenon of splitting the spectrum of the reflected signal from fiber-optic sensors (FOS) based on fiber Bragg gratings (FBG), due to which errors in calculating the strain values in diagnostic systems of the stress-strain state occur. The physical principles of splitting and their influence on the diagnostic system as a whole are considered. Various algorithms are proposed to solve the splitting problem, followed by their comparison. The optimal algorithm for processing the digitized optical spectra of signals from FOS was estimated using a mathematical model using real strain indicators from a bench experiment. FOS deformation data were obtained during an experiment on a bench for bending metal levers and were compared with strain gauges.

Dividing Splittable Goods Evenly and With Limited Fragmentation

A splittable good provided in n pieces shall be divided as evenly as possible among m agents, where every agent can take shares from at most F pieces. We call F the fragmentation and mainly restrict attention to the cases $$F=1$$F=1 and $$F=2$$F=2. For $$F=1$$F=1, the max–min and min–max problems are solvable in linear time. The case $$F=2$$F=2 has neat formulations and structural characterizations in terms of weighted graphs. First we focus on perfectly balanced solutions. While the problem is strongly NP-hard in general, it can be solved in linear time if $$m\ge n-1$$m≥n-1, and a solution always exists in this case, in contrast to $$F=1$$F=1. Moreover, the problem is fixed-parameter tractable in the parameter $$2m-n$$2m-n. (Note that this parameter measures the number of agents above the trivial threshold $$m=n/2$$m=n/2.) The structural results suggest another related problem where unsplittable items shall be assigned to subsets so as to balance the average sizes (rather than the total sizes) in these subsets. We give an approximation-preserving reduction from our original splitting problem with fragmentation $$F=2$$F=2 to this averaging problem, and some approximation results in cases when m is close to either n or n / 2.

Simultaneous elastic shape optimization for a domain splitting in bone tissue engineering

This paper deals with the simultaneous optimization of a subset O 0 of some domain Ω and its complement O 1 = Ω ∖ O 0 ¯ both considered as separate elastic objects subject to a set of loading scenarios. If one asks for a configuration which minimizes the maximal elastic cost functional both phases compete for space since elastic shapes usually get mechanically more stable when being enlarged. Such a problem arises in biomechanics where a bioresorbable polymer scaffold is implanted in place of lost bone tissue and in a regeneration phase, new bone tissue grows in the scaffold complement via osteogenesis. In fact, the polymer scaffold should be mechanically stable to bear loading in the early stage regeneration phase and at the same time, the new bone tissue grown in the complement of this scaffold should as well bear the loading. Here, this optimal subdomain splitting problem with appropriate elastic cost functionals is introduced and the existence of optimal two-phase configurations is established for a regularized formulation. Furthermore, based on a phase-field approximation, a finite-element discretization is derived. Numerical experiments are presented for the design of optimal periodic scaffold microstructure.

An optical solution for the set splitting problem

We describe here an optical device, based on time-delays, for solving the set splitting problem which is well-known NP-complete problem. The device has a graph-like structure and the light is traversing it from a start node to a destination node. All possible (potential) paths in the graph are generated and at the destination we will check which one satisfies completely the problem's constrains.