# Sigmoidal Mathematical Models in the Planning and Control of Rigid Pavement Works
## 4 Results
### Validation of Sigmoidal Models  
Figure 10 presents a comparison of the four models on the cost-time data corresponding to 28 rigid paving projects. These projects were used to validate the models. This evaluation enabled the analysis of the robustness, flexibility, and representability of each model under various empirical conditions. All models demonstrated a high degree of correlation between the observed data and the experimental results, following the characteristic sigmoid shape of the behavior of the accumulated cost as a function of time. However,  
discrepancies were identified in terms of local accuracy during the initial and accelerated transition phases of the construction process. The proposed model, based on a modified logistic function, demonstrated remarkable performance in most cases. Its structural flexibility enabled the accurate capture of intermediate growth patterns, a notable feature that distinguishes it for its capacity to represent empirical trajectories with smooth and symmetrical transitions. The logistic model demonstrated an adequate fit in global terms, adequately representing the displaced inflection points or curves with greater asymmetry, thereby enhancing accuracy in projects where the construction rhythm changed abruptly between phases. The Von Bertalanffy model has been demonstrated to be a suitable representation of asymmetric behavior, characterized by gradual initiation and a swift escalation in expenses during the intermediate stages. This model demonstrated particular efficacy in projects characterized by a transition in construction dynamics to a strategy of progressively increasing yields, subsequently followed by a phase of stabilization. The Gompertz model demonstrated particular efficacy in projects characterized by controlled construction progression, effectively representing gradual growth with an extended transition phase. However, its structural design tended to underestimate costs during intermediate phases in certain instances, particularly when there was a substantial acceleration in construction. The four models offered acceptable results; the proposed model achieved an adequate balance between fit, simplicity, and adaptability, which positioned it as the most  
Figure 10: Application of the different sigmoidal models for rigid pavement planning. robust alternative to represent the temporal evolution of cost in the evaluated projects. The logistic model and Gompertz, on the other hand, were the most effective for asymmetric growth scenarios, while the proposed model and Von Bertalanfry presented a more conservative performance. The parameters a, b, and c (see Table 4) were employed to obtain the models during training. These constants are utilized to create the mathematical model via an equation that can be employed in the execution of different rigid pavement projects. The factors that influence these elements include the efficiency of project execution, the availability of resources, and the complexity of the initial phases of the work. The project schedule and planning are also important factors, as they allow for efficient cash flow management and ensure that funds are available when costs are expected to be higher. Table 5 shows the statistics of the different models, which presented high levels of fit, with R values above 0.991 and determination coefficients (R\({}^{2}\)) above 0.982. These figures indicated a strong correlation between the predicted and observed values, which reflected an adequate predictive capacity for the evolution of cost over time. The logistic model obtained the highest value of R (0.9924) and R\({}^{2}\) (0.9848), followed by the Von Bertalanfry model (R = 0.9923), the Gompertz model (R = 0.9917), and the proposed model (R = 0.9910). The proposed model and the logistic model exhibited the lowest Root Mean Square Error (RMSE) value of 0.0506, thereby substantiating their superiority in the estimation of cost values.