Metaheuristics for Structural Design and Analysis. Yusuf Cengiz Toklu

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      Introduction

      I.1. Generalities

      Everything in our universe is related to some kind of optimization, minimization in losses and expenditures and maximization in gains. This is true for everything, from a stone or flowing water looking for the minimum potential energy position to living organisms trying to find the best solution when they come across a problem. As we all know from the laws of life, evolution is always towards the best fit, the word “best” pointing again to an optimization process.

      In human life also, whether performed using their own intelligence or by artificial intelligence developed by them, optimization is everywhere: from engineering design to construction planning, from personal economics to world economics, from transportation to water supply, from space research to deep-sea analysis, from self-care activities to organization of hospitals, from art activities to educational systems.

      Optimization is sometimes influenced by limited financial, physical and timely resources, sometimes influenced by certain intangible motivations like aesthetics and desire, and, most of the time, by a multitude of reasons. In real life, optimization concerning one single objective is very rare compared to multi-objective optimization. While performing optimization, we are usually bound by certain restrictions which are the constraints of the problem: the number of machines to be used in a production process, the magnitude of gravitational force that a pilot can withstand, or the strength limit of a given material. Thus, a general optimization problem can be formulated as

      [I.1]

      [I.2]

      where F(x) is a set of functions to be minimized called objective functions, g(x) is a set of equality constraints, h(x) is a set of inequality constraints and x is the set of unknowns over which all functions and constraints are defined. Optimization here is shown as minimization without losing generality, since a maximization function can be turned into a minimization function by just a simple multiplication with -1. The same is true between “smaller than” and “greater than” conditions.

      There have been very elegant techniques using classical tools of mathematics for solving this optimization problem, in cases where functions in F(x) are well defined and differentiable. Unfortunately, all of these methods are valid only in their range of applicability, as in linear programming, nonlinear programming, integer programming, gradient methods, etc. (Nocedal and Wright 2006; Fletcher 2013). In real life, most of the objective functions are such that they cannot be written down as a mathematical function, let alone a differentiable one. On the other hand, the unknowns may be any type of quantity, floating or integer numbers, names, directional expressions, the order of some activities, etc. Therefore, we can surely conclude that mathematical optimization methods cannot handle these problems, which form the great majority of problems encountered in real life.

      Metaheuristic methods, fortunately, are capable of handling all of these problems. Properly designed, they can help in making decisions on the best topology of a structural element, an economic activity with maximum income, the best hourly schedule in a school, the best route to follow between two points, etc. The term “metaheuristic” was first coined for the tabu search method, viewing it as “a metaheuristic superimposed on another heuristic” (Glover 1986). Humans started to use these techniques consciously in the 20th century, although nature was probably using them from the very beginning. For instance, evolutionary theory shows that living organisms of today started from single-celled beings, following the optimization rule СКАЧАТЬ