# Sigmoidal Mathematical Models in the Planning and Control of Rigid Pavement Works
## 6 Conclusions  
The findings of the research have demonstrated the applicability of sigmoidal models as effective tools for planning and progress control in rigid pavement projects. The findings indicate that the logistic and Von Bertalanfry models demonstrate optimal levels of fit, attaining coefficients of determination of 0.9848 and 0.9844, respectively. These models demonstrated a high degree of accuracy in representing the cumulative evolution of cost as a function of time, exhibiting satisfactory adaptability to the typical characteristics of S-curves. The proposed models not only represent the dynamic behavior in construction planning but also generate a sigmoidal envelope composed of the earliest and latest time curves. This approach establishes an early warning system that identifies discrepancies between actual and planned progress, enabling the technical team to make timely corrective decisions. The presence of actual values below the curve signifies delays, while those above reflect accelerated execution. Furthermore, the models' capacity to delineate upper and lower limits furnishes a quantitative framework for reference, thereby enhancing the monitoring of project performance and facilitating the optimization of schedule and resource management. In construction contexts characterized by elevated uncertainty or ambiguous objectives, these models have demonstrated their efficacy as robust instruments during the planning and execution phases. In terms of performance, the Von Bertalanffy model was effective in stable environments, while the Gompertz model and the proposed model showed adequacy in scenarios with asymmetric growth and controlled acceleration, so that the proposed model is effectively adapted to large-scale projects since its symmetry and lower error allow it to obtain greater reliability. This methodological versatility allows the analysis to be adapted to different types of projects. On the other hand, the logistic model, since it presents less dispersion, is a preferred option for general applications, opting for the different models are applied depending on the case of the project. The sigmoidal models used in this study allow accurate modeling of the time evolution of the cost, constituting a solid methodological basis for the development of intelligent control and scheduling systems in the field of road construction engineering. While the findings of this study demonstrate the efficacy of the proposed flexible sigmoidal model in representing the physical-financial progression in rigid pavement projects, the model necessitates precise and consistent historical data for its calibration. This may present a challenge in contexts where detailed or structured records are not readily available. Furthermore, sigmoidal models exhibit sensitivity to the presence of inconsistent or atypical data, which can adversely impact their predictive capability if rigorous debugging and validation processes are not implemented. In light of the aforementioned limitations, subsequent research endeavors will entail the incorporation of the proposed sigmoidal models within Building Information Modeling (BIM) environments, with a particular focus on 5D platforms that consider the interplay between the dimensions of time and cost. The integration would facilitate direct linkage of the estimates of physical-financial progress with the components of the three-dimensional digital model. This linkage would, in turn, allow for the simulation of different construction scenarios, the continuous evaluation of deviations from the planned schedule and budget, and the strengthening of support for operational decision-making in real time. This methodology, which integrates mathematical modeling and digital tools, signifies a substantial advancement toward a more efficient, automated, and transparent management of road infrastructure projects, in accordance with the principles of engineering 4.0. Conceptualization, J.M.P.O. and B.A.C.C. ; methodology, J.M.P.O. ; software, J.M.P.O. ; validation, M.E.M.P., J.L.P.T. and M.A.M.S.