## 3 The Multi-Objective Optimization Problem  
The optimal shape that minimizes both mass and tensile stresses of the voided beam is herein searched for. So, a MOOP is solved in order to find the best solution. Moreover, it is imposed that tensile stresses do not exceed the concrete cracking limit, given by (Eurocode 2): \(f_{ctm}=0.3f_{ck}^{2/3}\). Therefore, the optimization problem can be formulated as follows:  
_Find the_:  
\[\min_{\{p_{i}\}\in\ P_{ad}}\ J_{1},J_{2\max}(P_{1},\ \ldots,\ P_{9}) \tag{24}\]  
_By imposing the constraint_  
\[J_{2 max}\leq f_{ctm} \tag{25}\]  
where \(P_{ad}\) is a set of admissible parameters and \(J_{1}\), \(J_{2\max}\) represent the mass and the maximum tensile stress to be minimized, respectively [25]. Depending on the parameters included in Equation (15), the shape variation allows a possible solution set included in the Pareto optimal front. In the optimization process, the solver is able to find the best shape and dimensions of the voids to allow a more homogeneous distribution of stresses within the structure. Optimization Results--Octopus Solver (HypE Reduction)  
The Pareto optimal front in MOOPs represents a set of solutions that are non-dominated while being the best of all possible solutions. It means that a single solution cannot be simultaneously superior to all the other ones concerning all objectives. As a single goal improves in MOOP, another will get worse. Therefore, each solution of the Pareto set includes at least one objective that is inferior to another solution in that Pareto set, although both are superior to others in the rest of the search space [26]. This condition implies the necessity to define which is the best among all the feasible conditions. Figure 17: Tension (blue) (Sig1-Val) and compression (red) (Sig2-Val) path in the voided beam. In the case under examination, the Octopus solver adopts the search algorithm SPEA-2 to reach the convergence to the best global solution. SPEA2 uses a regular population and an archive. Starting with an initial population and an empty archive, the following steps are performed per iteration. First, all non-dominated population members are copied to the archive; any dominated individuals or duplicates are removed from the archive during this update operation. If the size of the updated archive exceeds a predefined limit, further archive members are deleted by a clustering technique that preserves the characteristics of the non-dominated front. Afterward, suitable fitness values are assigned to both archive and population members. The next step represents the mating selection phase where individuals from the union of population and archive are selected by means of binary tournaments. Finally, after recombination and mutation, the old population is replaced by the resulting offspring population [27]. After 250 generations, the solver selected a set of solutions, as shown the Figure 18. Three possible configurations are considered, in which the first solution (Test 1) shows the smallest values of tensile stresses among all feasible solutions, the third solution (Test 3) is the solution with the most negligible mass, and the second solution (Test 2) is characterized by mid-values with respect to the two edge solutions. The results after optimization (Figure 19), compared with the FEA of the solid beam, are summarized in Table 2. Considering the solid beam as a comparative test case, Table 3 summarizes the percentage variations of the chosen solutions. Figure 18: Pareto optimal front; in red, all feasible solutions. By observing, it emerges that the enhancement of one objective function corresponds to a worsening of the other. The following graph (Figure 20) shows the trend of the objective functions: it is worth noting that as the mass increases, the maximum tensile stress decreases. Considering that concrete is characterized by good behavior in compression but a bad one in traction, for the purposes of this study, the most performing solution is given by Test Case 1.