# Intelligent Total Transportation Management System for Future Smart Cities
## 2 Materials and Methods
### General Optimization Method  
At first sight, the required optimization methods can be defined simply as a nonlinear optimization problem. The mathematical representation can be given in a simple form:  
\[ \begin{split}&\text{\emph{min}}f(\mathbf{x})\\ &\text{\emph{subject to}}:\ g_{i}(\mathbf{x})\leq 0\ \ \text{for each}\ i\in\{1,\ 2,\ \ldots,\ n\}\\ &\text{\emph{and}}\ h_{j}(\mathbf{x})=0\ \ \text{for each}\ j\in\{1,\ 2,\ \ldots,\ m\}\\ &\text{\emph{where}}\ \mathbf{x}\in X\end{split}  \tag{1}\]  
where \(\mathbf{x}\): the vector of variables, \(f(\mathbf{x})\): objective function, \(g_{i}(\mathbf{x})\): non-equality constraint functions, \(h_{j}(\mathbf{x})\): equality constraint functions, and \(X\): space of the variables. The formulation of this optimization problem dealing with a single total (overall) transportation system seems to be too ambitious. Millions of solutions to such optimization problems are reduced to a part of the transportation systems, to a given type of sub-systems. The main reason for reducing the problem is the use of a fully and well observable system. However, the latest results of sciences and technologies allow solving the optimization of the total transportation system. For example, the Internet of Things, big data analysis, and soft computing make it possible to estimate the sub-systems of transportation being composed of non-cooperative or semi-cooperative vehicles (people) from the external information provided, e.g., by smart cards or video surveillance. The most important novelties of the introduced optimization problem are not related to the focus of a single transportation system and not to the optimization of the total system. The novelties are caused by the classification of the transportation system. As described in Section 2.2, there are horizontal and hierarchical classifications of the total transportation system. This classification allows us to define special objective functions for a part of the total transportation system and introduce more constraints. By adapting the total impact concept [39; 46; 47], the objective function can be defined in several different forms depending on the management's objectives. One of the most important primary objectives is the minimization of the energy used by transportation. Total energy consumption, \(E_{t}\), used by total urban transport system related an hour or day (or season) can be defined as:  
\[ \begin{split} E_{t}=f(x)=\frac{1}{\tau}\sum_{i=1}^{N}\int_{t_{i \geq 0}}^{t_{i\leq 5}}w_{\tau_{i}}e_{vc_{i}}\Big{(}\mathbf{r}_{v_{i}}, \mathbf{p}_{v_{i}},\mathbf{p}_{vd_{i}},\mathbf{o}_{v_{i}},\mathbf{x}_{v_{i}}, \mathbf{z}_{v_{i}},\ldots,\ \xi_{v_{i}},\ t\Big{)}dt,\\ \mathbf{x}^{T}=\Big{[}\mathbf{r}_{v_{i}},\mathbf{p}_{v_{i}}, \mathbf{p}_{v_{i}d_{i}},\mathbf{o}_{v_{i}},\mathbf{x}_{v_{i}},\mathbf{z}_{v_ {i}},\ldots\Big{]}\in X\end{split}  \tag{2}\]  
where \(\tau\) is the time, frame of reference; \(i=1\), \(2\),.... \(N\) number of vehicles; **w, e, r, p, o, x, z, \(t\)** are the weighting coefficient, energy consumption, real pathway, parameters, operational characteristics, vehicle motion characteristics/performance, environmental characteristics, noise vector, and time; while the indexes \(t_{s_{i}\geq 0}\), \(t_{f\leq t}\) mean motion starting and finishing times (forgiven \(i\)-th vehicle); **v**--given vehicle (\(v_{i}\) depicts the \(i\)-th vehicle); _vc, vd_ are related to the vehicle instantaneous consumption and a human vehicle driver. Here, \(\mathbf{r}_{v_{i}}\) vector characterizes the real pathway (slopes, curves of the road, track) along which the given \(i\)-th vehicle moves during \([t_{s_{i}\geq 0},\ t_{f\leq t}]\) time period. Vectors \(\mathbf{p}_{v_{i}},\mathbf{p}_{vd_{i}}\) are completed from the vehicle (types, sizes, empty mass, engine, engine performance) and driver characteristics (dynamics, reaction time).