## 4 Multi-Parametric MINLP Optimization
### Typical MINLP Optimization  
This paper presents the typical MINLP optimization of a 10 m long simply supported CFS that has been subjected to self-weight and uniformly distributed variable-imposed load of 6 kN/m\({}^{2}\).  
The superstructure of the considered composite floor structure with welded I sections (CFSWIS) comprises 3 different classes of steel, 7 classes of concrete, 81 alternative sizes of welded I sections, 25 alternative sizes of reinforcing meshes, 27 alternatives of plate depths, and 2 alternative positions of the centroid axis, resulting in 2.296 \(\times\) 10\({}^{6}\) different structural alternatives (structures) as a combination between the mentioned alternatives. Similarly, 5.103 \(\times\) 10\({}^{5}\) different structure alternatives are defined for the composite floor structure with standard IPE profiles (CFSIPE) and 6.804 \(\times\) 10\({}^{5}\) different alternatives for the composite floor structure with standard HEA profiles (CFSHEA). A total of 3.486 \(\times\) 10\({}^{6}\) different structural alternatives for all three structure types is summed up. One structural alternative is the optimal structure (for the defined objective function).  
Note that 15 individual MINLP optimizations of the composite structure have to be performed for 5 different objective functions (the mass, the direct cost, and three different energy life-cycle cost functions) in the combination with 3 different types of steel I beams (welded, IPE and HEA sections).  
The obtained optimal results are presented in Table 2 and Figure 5. While welded I sections are in the mass optimization and in the direct cost optimization calculated to be the most suitable steel beams, all three types of energy life-cycle cost optimizations exhibit IPE profiles to be optimal steel sections. The mass calculated in the direct cost optimization is about 2.5-times higher than the optimal mass optimized in the mass optimization. It should be noted that the worse the energy consumption case (from EEC to HEC) is, the lower the amount of calculated mass is. The direct costs calculated in the energy life-cycle cost optimization (for the high energy consumption case ELCC HEC) and in the mass optimization are 20% and 30% higher, respectively, than the optimal direct costs calculated in the direct cost optimization.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Objective Type** & **Mass kg/m\({}^{2}\)** & **DC EUR/m\({}^{2}\)** & **ELCC EUR/m\({}^{2}\)** \\ \hline MASS objective & **169.38** & 80.90 & - \\ DC objective & 422.97 & **62.29** & - \\ ELCC EEC objective & 288.63 & 63.64 & **91.81** \\ ELCC EIC objective & 217.42 & 65.70 & 118.91 \\ ELCC HEC objective & 175.45 & 74.67 & 170.38 \\ \hline \hline \end{tabular} Legend: DC, direct costs; ELCC, energy life-cycle costs; EEC, energy efficient case; EIC, energy inefficient case; HEC, high energy consumption case.  
\end{table}
Table 2: The optimal results.  
Comparative Diagrams of Competitive Spans of Composite Floor Structures for Different Objective Functions  
After the MINLP calculations were carried out, the computed results and the composite designs were analyzed and compared. First, comparative diagrams were drawn to determine the competitive spans of the CFSs with welded, IPE, and HEA profiles for different objective functions.