# Sigmoidal Mathematical Models in the Planning and Control of Rigid Pavement Works
## 5 Discussion  
The research addressed a critical factor in the impediment of rigid pavement works in Peru: deficient planning. In this context, sigmoidal models were employed, which have demonstrated efficacy in the scheduling of construction projects. [PERSON] and [PERSON] [18] demonstrated that their model exhibited an adequate fit in 75% of the projects that were analyzed. In a similar vein, [PERSON] et al. [37] employed fuzzy mathematical models to ascertain construction schedules and assess contingencies engendered by schedule compression and delays due to material shortages that were not foreseen. Their findings underscore the significance of prioritizing activities with minimal residual capacity for adjustment, as opposed to the mere allocation of materials to activities that are prepared to commence immediately. This finding lends credence to the hypothesis that mathematical models can be utilized to represent common patterns in construction planning. Figure 14: The graphical interface of the Logistics Model in Dynamo Player. Figure 15: Project S-curve. The database used in this research consisted of 140 rigid pavement projects in Peru, associated with local regulations, construction practices, availability of materials, and particular climatic conditions with cost and time variables for training the sigmoidal models. This is similar to that presented by [PERSON] and [PERSON] [14], who used 101 real projects to develop a predictive model based on four factors: contract amount, duration, type of work and location, using these as input variables to estimate the S-curve parameters and thus proposed a cubic polynomial function, which showed advantages in both accuracy and simplicity over traditional formulas. This geographical and sectoral restriction limits the extrapolation of the models to other types of infrastructure or regions with different construction dynamics. Furthermore, the data under consideration primarily concentrated on cost and duration variables, neglecting to incorporate qualitative factors such as the level of technology employed, contractor experience, or technical complexity. These qualitative factors could have a substantial impact on the temporal behavior of physical progress. To address the variability of cost data due to temporal and geographic differences, a statistical normalization process was applied. This process standardized all cost values within a range from 0 to 1. This transformation allowed for the elimination of the direct influence of monetary units and absolute disparities in the values recorded. Consequently, comparability between projects executed in different geographic and economic contexts was fostered. By focusing on the ratio of cumulative progress to total project cost, the model can capture the relative behavior of financial progress without being affected by macroeconomic factors such as inflation or regional differences in material or labor prices. To reduce the gaps in the research literature, an alternative approach based on sigmoidal mathematical models, specifically adapted to S-curves, was proposed. This adaptation allowed the construction of a more flexible model adjusted to the real behavior observed on site, especially in the planning of road projects. The model achieved R\({}^{2}\) higher than 98%, exceeding the performance of classical models, which coincides with [PERSON] et al. [5], who pointed out that, over time, multiple mathematical formulations have been developed to estimate S-curves in construction projects, including polynomial functions, exponential and transformational approaches, which are limited and not applicable to road works projects. In comparison with previous models, such as the support vector machine model presented by [PERSON] et al. [7], our model achieved an R\({}^{2}\) of 0.94 and an MAE of 3.6000, which contrasts significantly with the proposed model, logistics model, Von Bertalanffy model, and Gompertz model in this research, which obtained a higher R\({}^{2}\) from 0.9821 to 0.9848 and a lower MAE from 0.0278 to 0.0321.