## 3 Results
### Elaborating the Best Fit of Cumulative Cost Curves  
Based on the specified area, the course of the curve with the best fit to the function was determined. Initially, polynomial regression (sixth-degree polynomial) and the trend function were used to describe the course of the cost curve. The use of a sixth-degree polynomial trend facilitated achieving a high correlation coefficient (close to unity). This proved the existence of a very strong correlation relationship and a very good description of the studied phenomenon. Moreover, a low coefficient of variation was achieved, indicating the consistent nature of the analyzed feature and the uniformity of the studied projects. However, applying higher-order polynomials (higher than a fourth degree) may prove \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Construction Group/Sector**} & **Actual Cost Curve that** & & **Actual Cost Curve that** \\  & **Limits the Area “Upper Bound Line”** & & **Actual Best-Fit Curve** \\ \hline Apartment houses (A) & y = \(-1.06x^{3}+1.19x^{2}+0.87\)x & y = \(-1.04x^{3}+1.72x^{2}+0.32\)x & y = \(0.09x^{3}+0.54x^{2}+0.37\)x \\ Office buildings (B) & y = \(-1.16x^{3}+1.49x^{2}+0.47\)x & y = \(-0.94x^{3}+1.64x^{2}+0.16\)x & y = \(0.56x^{3}+0.23x^{2}+0.15\)x \\ Hotel buildings (C) & y = \(-1.20x^{3}+2.01x^{2}+0.19\)x & y = \(-0.60x^{3}+1.59x^{2}+0.01\)x & y = \(0.94x^{3}+0.04x^{2}+0.02\)x \\ Commercial and service & y = \(-1.30x^{3}+1.99x^{2}+0.31\)x & y = \(-0.77x^{3}+1.56x^{2}+0.21\)x & y = \(0.94x^{3}+2.34x^{2}-0.40\)x \\ buildings (D) & y = \(-1.26x^{3}-1.67x^{2}+1.45\)x & y = \(-0.12x^{3}+0.11x^{2}-0.07\)x & y = \(-0.01x^{3}+0.06x^{2}-0.01\)x \\ \hline All buildings & y = \(-1.20x^{3}+1.40x^{2}+0.82\)x & y = \(-0.67x^{3}+1.36x^{2}+0.31\)x & y = \(0.86x^{3}+0.10x^{2}+0.04\)x \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Envelope of the actual cost curve—fragment of the results. overly complex for practical decision-makers such as investors and contractors. Therefore, further scientific research aimed at identifying simpler mathematical models or formulas was pursued to accurately represent the cumulative cost curve. It was therefore considered reasonable to use a third-degree polynomial in order to describe the best fit of the cost curve. The shape of the cumulative cost curve was analyzed, noting its initial and final flat segments during the construction project. This is attributed to the gradual commencement and completion of the project. Initially, activities involve organizing human resources, finalizing contracts with contractors and subcontractors, preparing the construction site, and conducting basic preparatory work. As time progresses, activities accelerate, as reflected in the curve's shape. Multiple workstations operate concurrently with specialized work brigades, leading to increased construction activity and costs. This phase contrasts sharply with the slower initial and final stages of implementation. Based on the analysis of both the literature on this subject and the shape of the cost curve for construction projects collected in the knowledge base, an attempt was made to best fit the cost curve using a third-degree polynomial. An S-shaped cost curve can be mathematically described by two convexities and one inflection point (\(x_{0}\)). The cost curve in the initial phase of construction (the first phase) exhibits convexity, geometrically meaning that the curve lies above its tangent at each point within the interval \(<0,x_{0}\)). As construction progresses and activities intensify over time, the cost curve steeply inclines relative to the time axis in the missile phase. The cost curve reaches an inflection point (\(x_{0}\)), signaling the transition to the second phase of implementation, where the rate of cost increase begins to decelerate. In the second phase of construction, the cost curve becomes concave, meaning it curves upwards; geometrically, this indicates that the curve lies below its tangent at each point within the interval \((x_{0},1>)\).