## 3 Methodology
### Model Formulation  
At the core of UTMOM is the objective function, a meticulously designed mathematical formula that aims to ensure the model's predictions closely match the actual observed data. This function evaluates the precision of UTMOM's predictions by minimizing discrepancies between actual vehicle counts and the model's estimates for each designated time slot and day across both datasets. As the cornerstone of UTMOM's optimization efforts, this objective function is essential in guiding the model toward generating the most accurate and reliable forecasts of urban traffic flows. * Objective Function: To represent the difference between two datasets, an objective function is designed to minimize the weighted sum of squares of these differences, carefully factoring in the significance of each observation through weights. In this context, \(w_{ijk}\) and \(w^{\prime}_{ijk}\) are the weights for the estimated vehicle counts in the first and second datasets, respectively, symbolized by \(\hat{x}_{ijk}\) and \(\hat{x}^{\prime}_{ijk}\). This minimization spans across all corresponding periods--morning, noon, and evening--over five days, aiming to refine the model's estimates by emphasizing the importance of aligning them closely with the observed data, considering the distinct weights for each dataset and time interval. The selection of weights in the objective function is crucial for accurately reflecting the importance of different observations. These weights are determined based on empirical data analysis and expert judgment, ensuring that the most critical time periods and conditions are given appropriate emphasis. For instance, higher weights are assigned to peak traffic times and areas with historically higher traffic variability, as discrepancies during these periods have a more significant impact on overall traffic management. \[\textit{Minimize }\sum_{i=1}^{5}\sum_{j=1}^{3}\sum_{k=1}^{2}\left(w_{ijk}\cdot\left( \hat{x}_{ijk}-D_{ij}\right)^{2}+w^{\prime}_{ijk}\cdot\left(\hat{x}^{\prime}_{ijk }-D^{\prime}_{ij}\right)^{2}\right) \tag{1}\]  
* Total Number of Vehicles Constraint: The sum of the vehicle estimates for both the first and second datasets should be close to the actual total number of vehicles. These constraints ensure that the total estimated vehicles for each period in both datasets are similar to the actual observed totals. These constraints are shown in Equations (2) and (3). \[\sum_{j=1}^{3}\hat{x}_{ijk}\ \approx\ actual\ totals\] (2) \[\sum_{j=1}^{3}\hat{x}^{\prime}_{ijk}\ \ \approx\ actual\ totals^{\prime}\] (3)
* Volume Constraints: Volume constraints serve to ensure that the total estimated number of vehicles across both \(k\) condition for each particular day \(i\) and time period \(j\) aligns with the number of vehicles recorded in the initial dataset. Formally, the constraints for each day \(i\) and each specified period \(j\) (e.g., morning, noon, evening) are represented mathematically in Equations (4) and (5). \[\sum_{k=1}^{2}\ \hat{x}_{ijk}=D_{ij},\quad\forall i,j\] (4) \[\sum_{k=1}^{2}\ \hat{x}_{ijk}=D^{\prime}_{ij},\quad\forall i,j\] (5)
* Minimum and Maximum Vehicle Number Constraints: The model incorporates explicit bounds on the estimated vehicle counts to maintain operational plausibility and prevent anomalous predictions. These constraints are formally defined in Equations (6) and (7). These constraints are applied universally across all intervals \(k\), periods \(j\), and days \(i\), thereby ensuring that the estimations do not exceed the maximum capacity expected under normal conditions nor fall below a reasonable threshold that could indicate underutilization or data collection errors. By imposing these restrictions, the model remains sensitive to the bounds of typical traffic conditions and avoids the propagation of extreme values that could skew analysis and decision-making processes.