*3.1. Formulation*

Optimization problems can be divided into two large groups in general—constrained and unconstrained problems. Because of the limitations required for the design of structures, the optimum design of RCRWs is a constrained optimization problem, which can be expressed as follows [25]:

$$\begin{array}{l}\text{Minimize } \mathbf{F}(\mathbf{x}) \text{ subject to } \operatorname{g}\_{\mathbf{i}}(\mathbf{x}) \le 0\\\mathbf{j} = 1, 2, \dots, m; \; \mathbf{x}\_{\mathbf{j}} \in \mathbf{R}^{\mathbf{d}}, \; \mathbf{j} = 1, 2, \dots, n\end{array} \tag{22}$$

in which *F*(*x*) is the objective function; *g*i(*x*) is the *i*-th constraint; *m* and *n* are the total number of constraints and design variables, respectively; *R*<sup>d</sup> is a given set of discrete values from which the individual design variables *x*j can take values. In this paper, an exterior penalty function method

was used to transform the constrained structural optimization problem into an unconstrained one, as follows [26]:

$$\Phi(\mathbf{x}, r\_{\rm P}) = F(\mathbf{x}) \left| 1 + r\_{\rm P} \sum\_{\mathbf{r}=1}^{\rm m} \left( \max \left( \frac{\mathcal{G}\mathbf{r}}{\mathcal{G}\_{\rm r, all}} - 1, 0 \right) \right) \right| \tag{23}$$

where φ is the pseudo (penalized) objective function; *g*r and *g*r,all are the *r*-th constraint and its allowable value, respectively; and *r*p is a positive penalty parameter, which in this study was assumed to be equal to 25.
