## 4 Multi-Parametric MINLP Optimization  
The investigation of the optimal design of CFSs involves the multi-parametric MINLP optimization of a combination of different design parameters and superstructure alternatives, such as:  
* Five alternative loads (2-10 kN/m\({}^{2}\), 2 kN/m\({}^{2}\) step);
* Ten alternative spans (5-50 m, 5 m step);
* Three alternative steel classes (S 235, S 275, S 355);  
* IPE 600); \(\odot\) Twenty-four alternative HEA sections (HEA 100
- HEA 1000);  
* Twenty-seven alternative slab depths in whole centimeters (4-30 cm);
* Two alternative centre of gravity axis positions for all-steel section: \(\odot\) In the concrete plate; \(\odot\) In the steel profile;
* Five different objective alternatives: \(\odot\) Mass objective function (MASS); \(\odot\) Direct cost objective function (DC); \(\odot\) Three energy life-cycle cost objective functions of three different energy consumption cases (ELCC): \(\blacksquare\) An energy efficient case (ELCC EEC, 50 kWh/m\({}^{2}\)); \(\blacksquare\) An energy inefficient case (ELCC EIC, 100 kWh/m\({}^{2}\));A high energy consumption case (ELCC HEC, 200 kWh/m2).  
Note that the discrete alternatives of the loads, spans, dimensions (profiles), material classes, and the positions of the centroid axis in this paper are the same as how they are they defined in reference [14], which allows a comparison between the results of the different objective functions that are used. In this way, the combinatorics of the multi-parametric study yields 1.743 \(\times\) 108 different structure alternatives. The defined optimization problem thus arises to the multi-parametric MINLP mass, direct cost, energy life-cycle cost, material, standard dimension, and rounded dimension optimization problem. The optimizations are performed using a MINLP program MIPSYN [39; 40]. The modified outer-approximation/equality-relaxation (OA/ER) algorithm by [PERSON] and [PERSON] [41], the successor of the OA/ER algorithm [42], is applied. The OA/ER algorithm solves an alternative sequence of non-linear programming (NLP) and mixed-integer linear programming (MILP) problems, as seen in Figure 4. The NLP performs continuous optimization with fixed binary variables (discrete alternatives) and represents an upper bound on the minimized MINLP objective function. The MILP calculates new binary variables (discrete alternatives) that involve a global linear approximation to the superstructure of discrete alternatives and predicts a lower bound on the minimized objective function. The NLPs and MILPs must be solved sequentially. Convergence is achieved when the MILP lower bound matches the NLP upper bound. The optimization of the non-convex problem is stopped if NLPs show no improvement.  
To solve the NLPs, we use GAMS/CONOPT (generalized reduced-gradient method) by [PERSON] [43], and to compute the MILP problems, we use GAMS/CPLEX (branch and bound method) [44]. To speed up the convergence of the modified OA/ER algorithm, a two-phase MINLP strategy [45; 46] is used. The two-phase strategy starts with the continuous NLP computation at the relaxed binary variables (discrete alternatives). The computed solution including the continuous variables provides a good starting point for the next discrete optimization. In the second phase, the binary variables (discrete alternatives) are recovered, and discrete MINLP optimization (the alternative sequence of NLPs and MILPs) is guided to the optimal result.  
The unit prices for manufacturing considered in the parametric optimization study are the same as those defined in reference [14]. The labour cost and steel price in high-income economies (20 EUR/h for labour and 1.25 EUR/kg for steel) are used in this study as they represent the average unit cost and price between middle-income and top-income  
Figure 4: Steps of the OA/ER algorithm.  
economies and because most composite structures are constructed in this region, see the World Bank [47] and reference [14].