Trim Loss Optimization by an Improved Differential Evolution

The “trim loss problem” (TLP) is one of the most challenging problems in context of optimization research. It aims at determining the optimal cutting pattern of a number of items of various lengths from a stock of standard size material to meet the customers’ demands that the wastage due to trim loss is minimized. The resulting mathematical model is highly nonconvex in nature accompanied with several constraints with added restrictions of binary variables. This prevents the application of conventional optimization methods. In this paper we use synergetic differential evolution (SDE) for the solution of this type of problems. Four hypothetical but relevant cases of trim loss problem arising in paper industry are taken for the experiment. The experimental results compared with those of the other techniques show the competence of the SDE to solve the problem

Trim Loss Optimization by an Improved Differential Evolution

The "trim loss problem" (TLP) is one of the most challenging problems in context of optimization research. It aims at determining the optimal cutting pattern of a number of items of various lengths from a stock of standard size material to meet the customers' demands that the wastage due to trim loss is minimized. The resulting mathematical model is highly nonconvex in nature accompanied with several constraints with added restrictions of binary variables. This prevents the application of conventional optimization methods. In this paper we use synergetic differential evolution (SDE) for the solution of this type of problems. Four hypothetical but relevant cases of trim loss problem arising in paper industry are taken for the experiment. The experimental results compared with those of the other techniques show the competence of the SDE to solve the problem

Arc flow formulations based on dynamic programming: Theoretical foundations and applications

Network flow formulations are among the most successful tools to solve optimization problems. Such formulations correspond to determining an optimal flow in a network. One particular class of network flow formulations is the arc flow, where variables represent flows on individual arcs of the network. For NP-hard problems, polynomial-sized arc flow models typically provide weak linear relaxations and may have too much symmetry to be efficient in practice. Instead, arc flow models with a pseudo-polynomial size usually provide strong relaxations and are efficient in practice. The interest in pseudo-polynomial arc flow formulations has grown considerably in the last twenty years, in which they have been used to solve many open instances of hard problems. A remarkable advantage of pseudo-polynomial arc flow models is the possibility to solve practical-sized instances directly by a Mixed Integer Linear Programming solver, avoiding the implementation of complex methods based on column generation. In this survey, we present theoretical foundations of pseudo-polynomial arc flow formulations, by showing a relation between their network and Dynamic Programming (DP). This relation allows a better understanding of the strength of these formulations, through a link with models obtained by Dantzig-Wolfe decomposition. The relation with DP also allows a new perspective to relate state-space relaxation methods for DP with arc flow models. We also present a dual point of view to contrast the linear relaxation of arc flow models with that of models based on paths and cycles. To conclude, we review the main solution methods and applications of arc flow models based on DP in several domains such as cutting, packing, scheduling, and routing

Numerical Analysis and Spanwise Shape Optimization for Finite Wings of Arbitrary Aspect Ratio

This work focuses on the development of efficient methods for wing shape optimization for morphing wing technologies. Existing wing shape optimization processes typically rely on computational fluid dynamics tools for aerodynamic analysis, but the computational cost of these tools makes optimization of all but the most basic problems intractable. In this work, we present a set of tools that can be used to efficiently explore the design spaces of morphing wings without reducing the fidelity of the results significantly. Specifically, this work discusses automatic differentiation of an aerodynamic analysis tool based on lifting line theory, a light-weight gradient-based optimization framework that provides a parallel function evaluation capability not found in similar frameworks, and a modification to the lifting line equations that makes the analysis method and optimization process suitable to wings of arbitrary aspect ratio. The toolset discussed is applied to several wing shape optimization problems. Additionally, a method for visualizing the design space of a morphing wing using this toolset is presented. As a result of this work, a light-weight wing shape optimization method is available for analysis of morphing wing designs that reduces the computational cost by several orders of magnitude over traditional methods without significantly reducing the accuracy of the results