Application of Robust Optimization to the Sawmill Planning Problem

Optimization models have been used to support decision making in the forest industry for a long time. However, several of those models are deterministic and do not address the variability that is present in some of the data. Robust Optimization is a methodology which can deal with the uncertainty or variability in optimization problems by computing a solution which is feasible for all possible scenarios of the data within a given uncertainty set. This paper presents the application of the Robust Optimization Methodology to a Sawmill Planning Problem. In the particular case of this problem, variability is assumed in the yield coefficients associated to the cutting patterns used. The main results show that the loss in the function objective value (the “Price of Robustness”), due to computing robust solutions, is not excessive. Moreover, the computed solutions remain feasible for a large proportion of randomly generated scenarios, and tend to preserve the structure of the nominal solution. We believe that these results provide an application area for Robust Optimization in which several source of uncertainty are present.     

On stabilizing volatile product returns

As input flows of secondary raw materials show high volatility and tend to behave in a chaotic way, the identification of the main drivers of the dynamic behavior of returns plays a crucial role. Based on a stylized production-recycling system consisting of a set of nonlinear difference equations, we explicitly derive parameter constellations where the system will or will not converge to its equilibrium. Using a constant elasticity of substitution production function, the model is then extended to enable coverage of real world situations. Using waste paper as a reference raw material, we empirically estimate the parameters of the system. By using these regression results, we are able to show that the equilibrium solution is a Lyapunov unstable saddle point. This implies that the system is sensitive on initial conditions that will hence impede the predictability of product returns. Small variations of production input proportions could however stabilize the whole system.     

On the complexity of project scheduling to minimize exposed time

We consider project scheduling where the project manager’s objective is to minimize the time from when an adversary discovers the project until the completion of the project. We analyze the complexity of the problem identifying both polynomially solvable and NP-hard versions of the problem. The complexity of the problem is seen to be dependent on the nature of renewable resource constraints, precedence constraints, and the ability to crash activities in the project.     

Geometric measures of convex sets and bounds on problem sensitivity and robustness for conic linear optimization

The effect of data perturbation and uncertainty has always been an important consideration in Optimization. It is important to know whether a given problem is very sensible to perturbations on the data or, on the contrary, is more “robust”. Problem geometry does have an impact on the sensitivity of the problem and in this paper we analyze this connection by developing bounds to the change in the optimal value of a conic linear problem in terms of some geometric measures related to the radius of inscribed and circumscribed balls to the feasible region of the problem. We also present a parametric analysis for Linear Programming which allows us to construct an estimate of safety limits for perturbations of the data. These results are developed in relation to questions in robust optimization.     

Identification of the lamination stack's behavior simulated with a multi-scale homogenization using a progressive contact formulation

The influence of the lamination stack on the mechanical behavior of an electrical motor is significant, especially for lightweight designs. Information about the stack's behavior is relevant for a proper calculation of the whole motor. This behavior is dependent on the contact between the single sheets resulting from the roughness and the behavior of the roughness peaks. The normal behavior is an elasto-plastic and progressive one, which results mainly of the number of roughness peaks in contact. Whereas the tangential micro-slip behavior is dependent on the single contacts and the friction coefficient. Both behaviors can be described by models and thus it is possible to calculate the behavior of a whole stack using an homogenization based on representative volume elements. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cichoń’s diagram,regularity properties and {\varvec{\Delta}^1_3} sets of reals

We study regularity properties related to Cohen, random, Laver, Miller and Sacks forcing, for sets of real numbers on the \({\varvec{\Delta}^1_3}\) level of the projective hieararchy. For \({\varvec{\Delta}^1_2}\) and \({\varvec{\Sigma}^1_2}\) sets, the relationships between these properties follows the pattern of the well-known Cichoń diagram for cardinal characteristics of the continuum. It is known that assuming suitable large cardinals, the same relationships lift to higher projective levels, but the questions become more challenging without such assumptions. Consequently, all our results are proved on the basis of ZFC alone or ZFC with an inaccessible cardinal. We also prove partial results concerning \({\varvec{\Sigma}^1_3}\) and \({\varvec{\Delta}^1_4}\) sets.     

Stabilization of mixture of two rigid solids modeling temperature and porosity

In this paper we investigate the asymptotic behavior of solutions to the initial boundary value problem for a mixture of two rigid solids modeling temperature and porosity. Our main result is to establish conditions which ensure the analyticity and the exponential stability of the corresponding semigroup.     

Drying of solids with irregular geometry: numerical study and application using a three-dimensional model

This article proposes a numerical solution for the diffusion equation applied to solids with arbitrary geometry using non-orthogonal structured grids for the boundary condition of the first kind. A transient three-dimensional mathematical formulation written in boundary fitted coordinates and numerical formalism to discretize the diffusion equation by using the finite volume method, including numerical analysis of the computational solution are presented. To validate the proposed solution, the results obtained in this work were compared with well-known numerical solution available in literature and good agreement was observed. In order to verify the potential of the proposed numerical solution, it was applied to describe mass transfer inside ceramic roof tiles during drying. For that, it was used experimental data of the drying kinetics at the following temperatures: 55.6; 69.7; 82.7 and 98.6 °C. An optimization technique using experimental dataset has been presented to estimation of transport properties. The obtained statistical indicators enable to conclude that the numerical solution satisfactorily describes the drying processes.     

Magnetic field tapering in the KfK coaxial gyrotron

Equations describing the magnetic field tapering in a gyrotron cavity with allowance for the dependence of the radius of the electron guiding centers on the longitudinal coordinate are derived. Influence of the shift of the magnet of the KfK coaxial gyrotron along the gyrotron axis onefficiency is studied. It is found that for the operating voltage 90kV the optimal alignment is such that the maximum of the magnetic field is placed in the middle of the straight section of the cavity. For 80kV the absolute values of the field must be reduced by factor 1.017, or the maximum of the field must be placed in the output taper of the cavity.