## 2 MINLP Optimization Models  
The problem of steel-concrete CFSs is a non-linear, continuous, and discrete optimization problem that can be solved by mixed-integer non-linear programming, i.e., MINLP. The general MINLP problem is formulated in the following way:  
\[\min z=f(\mathbf{x},\mathbf{y})\]  
subjected to:  
\[\mathbf{g}(\mathbf{x},\mathbf{y})\leq\mathbf{0}\;(\text{MINLP})\] \[\mathbf{x}\;\in\;X=\big{\{}\mathbf{x}\;\in\;R^{n}:\;\mathbf{x}^{ \text{LO}}\;\leq\;\mathbf{x}\;\leq\;\mathbf{x}^{\text{UP}}\big{\}}\] \[\mathbf{y}\;\in\;Y\;=\;\{0,l\}^{\text{m}}\]  
where \(\mathbf{x}\) and \(\mathbf{y}\) are the vectors of the continuous and discrete binary (0-1) variables, respectively. \(f(\mathbf{x},\mathbf{y})\) is the objective function subjected to the (in)equality constraints \(\mathbf{g}(\mathbf{x},\mathbf{y})\leq\mathbf{0}\). Functions \(f(\mathbf{x},\mathbf{y})\) and \(\mathbf{g}(\mathbf{x},\mathbf{y})\) must be continuous and differentiable; at least one of them must be non-linear. \begin{table}
\begin{tabular}{l l} \hline \hline
**Abbreviation** & **Meaning** \\ \hline DC & direct production costs per unit of useable surface of the structure \\ CFS & composite floor structure (system) \\ CFSHEA & composite floor structure with HEA sections \\ CFSIPE & composite floor structure with IPE sections \\ CFSWIS & composite floor structure with welded I sections \\ EEC & energy efficient case \\ ELC & energy life-cycle costs of the structure per unit of its useable surface \\ ELCC EEC & energy life-cycle cost criteria/optimization of an energy efficient case \\ ELCC EIC & energy life-cycle cost criteria/optimization of an energy inefficient case \\ ELCC HEC & energy life-cycle cost criteria/optimization of a high energy consumption case \\ GAMS & general algebraic modelling system \\ HEA & European wide flange section \\ HEC & high energy consumption case \\ HVAC & heating, ventilation and air conditioning \\ IPE & I profile European (European IPE section) \\ MASS & mass of the structure per unit of its useable surface \\ MINLP & mixed-integer non-linear programming \\ MILP & mixed-integer linear programming \\ NLP & non-linear programming \\ OA/ER & outer-approximation/equality-relaxation algorithm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of abbreviations. For the optimization of a composite I beam floor system, see Figure 2, where a number of various optimization models were modelled according to the above MINLP formulation. These models, which were already presented by [PERSON] et al. [14], are now extended to three different criteria--three different objective functions that are necessary for structures, i.e., minimization of mass, direct costs, and energy life-cycle costs. The defined objective functions are subjected to design, loading, and dimensioning constraints. The latter are determined according to Eurocode [31; 32; 33; 34; 35] specifications by taking into account the (in)equality conditions for the ultimate limit state constraints (internal forces, resistances) and the serviceability limit state constraints (deflections). In a previous study [14], which discussed the plastic and elastic resistances of composite structures and three various positions of the neutral axes (in the web of the steel I beam, in the upper steel flange and in the concrete), it was found that the CFSs calculated with plastic resistance are 15% cheaper than those with elastic resistance when HEA and IPE profiles for steel beams are used and that they are 20% cheaper when welded I-beams are used. It was found that the composite constructions with the plastic resistance and the neutral axis in the plate are the most favourable constructions. For this reason, only the plastic resistance of the composite structures and the neutral axis located in the concrete slab are considered in this study, as seen in Figure 3.  
Note that the optimization models are modelled in the GAMS (general algebraic modelling system) by [PERSON] et al. [36].