Models for capacitated lot-sizing problem with backlogging,setup carryover and crossover

Setup operations are significant in some production environments. It is mandatory that their production plans consider some features, as setup state conservation across periods through setup carryover and crossover. The modelling of setup crossover allows more flexible decisions and is essential for problems with long setup times. This paper proposes two models for the capacitated lot-sizing problem with backlogging and setup carryover and crossover. The first is in line with other models from the literature, whereas the second considers a disaggregated setup variable, which tracks the starting and completion times of the setup operation. This innovative approach permits a more compact formulation. Computational results show that the proposed models have outperformed other state-of-the-art formulation.     

Stability and convergence of the spectral Lagrange-Galerkin method for mixed periodic/non-periodic convection-dominated diffusion problems

We present a convergence analysis of the spectral Lagrange-Galerkinmethod for mixed periodic/non-periodic convection-diffusionproblems. The scheme is unconditionally stable, independentof the diffusion coefficient, even in the case when numericalquadrature is used. The theoretical predictions are illustratedby a series of numerical experiments. For the periodic case,our results present a significant improvement on those givenby Süli & Ware (1991) SIAM J. Numer.Anal.28, 423-445).     

Rates of water exchange on the [Fe4(OH)2(hpdta)2(H2O)4]0 molecule and its implications for geochemistry

The ammonium salt of [Fe(4)O(OH)(hpdta)(2)(H(2)O)(4)](-) is soluble and makes a monospecific solution of [Fe(4)(OH)(2)(hpdta)(2)(H(2)O)(4)](0)(aq) in acidic solutions (hpdta = 2-hydroxypropane-1,3-diamino-N,N,N',N'-tetraacetate). This tetramer is a diprotic acid with pK(a)(1) estimated at 5.7 ± 0.2 and pK(a)(2) = 8.8(5) ± 0.2. In the pH region below pK(a)(1), the molecule is stable in solution and (17)O NMR line widths can be interpreted using the Swift-Connick equations to acquire rates of ligand substitution at the four isolated bound water sites. Averaging five measurements at pH < 5, where contribution from the less-reactive conjugate base are minimal, we estimate: k(ex)(298) = 8.1 (±2.6) × 10(5) s(-1), ΔH(++) = 46 (±4.6) kJ mol(-1), ΔS(++) = 22 (±18) J mol(-1) K(-1), and ΔV(++) = +1.85 (±0.2) cm(3) mol(-1) for waters bound to the fully protonated, neutral molecule. Regressing the experimental rate coefficients versus 1/[H(+)] to account for the small pH variation in rate yields a similar value of k(ex)(298) = 8.3 (±0.8) × 10(5) s(-1). These rates are ～10(4) times faster than those of the [Fe(OH(2))(6)](3+) ion (k(ex)(298) = 1.6 × 10(2) s(-1)) but are about an order of magnitude slower than other studied aminocarboxylate complexes, although these complexes have seven-coordinated Fe(III), not six as in the [Fe(4)(OH)(2)(hpdta)(2)(H(2)O)(4)](0)(aq) molecule. As pH approaches pK(a1), the rates decrease and a compensatory relation is evident between the experimental ΔH(++) and ΔS(++) values. Such variation cannot be caused by enthalpy from the deprotonation reaction and is not well understood. A correlation between bond lengths and the logarithm of k(ex)(298) is geochemically important because it could be used to estimate rate coefficients for geochemical materials for which only DFT calculations are possible. This molecule is the only neutral, oxo-bridged Fe(III) multimer for which rate data are available.