Optimizing the Cross Section of Cold-Rolled Steel Beams Using a Genetic Algorithm: Avoiding Local Optima Using Adaptive Mutation Control, Flexible Restriction Handling and Inbreed Avoiding Mating Strategies

Abstract

In modern mechanical engineering and steelwork the use of cold-rolled steel sections is a standard method. These sections should be mechanically stable on the one hand and cost efficient on the other hand. To decide what profile suits for a certain case is a constrained optimization problem which is in general non convex, i.e. several local optima exist. To solve this non trivial problem we used genetic algorithms, search heuristics that mimic the process of natural evolution. For the specific application some additional problems had to be solved: First, an adaptive mutation control was implemented. Second, a mixed asexual and sexual reproduction was applied with an inbreed avoiding method based on the genetic distance of the individuals. Third, the restrictions were handled flexible, dependent on the mutation strength. This means that under the conditions of strong mutations (r-strategy), violations of the restrictions are allowed within some limits corresponding to reduced evolutionary pressure. Later on when approaching an optimum and the algorithm changes eventually to K-strategy, the restrictions become more severe corresponding to stabilising selection. The presented algorithm was tested on some cases; we found that significant improvement of cost efficiency was reached while mechanical stability was still granted. In comparison to hard restriction implementations like constant penalty functions or Lagrange-multipliers due to the flexible restrictions the algorithm tends significantly less to sustain in local optima. This approach could help to find cost efficient and light weight steel structures for mechanical engineering in the near future. case. Usually, the required mechanical parameters are calculated from the loads expected. Then either the cheapest profile from a palette of standard products is selected just fulfilling the mechanical requirements (including some safety-factor) or some experienced engineer uses inspiration and perspiration to design a profile where the material is "optimally" exploited. The optimization problem could be solved using a method from the field of nonlinear programming, where the problem is defined by a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are nonlinear [2]. Formally: Let n, m, and p be positive integers. Let X be a subset of ℝ n , let f, g i , and h j be real-valued functions on X for each i in {1, …, m} and each j in {1, …, p}.