# Multiagency Modeling of Transformation Strategies Towards Sustainable Smart Cities
## 3 The Multiagency Sustainable Smart City Model and Transformation Strategies
### Model Description
\begin{table}
\begin{tabular}{c c} \hline \(D_{\rm i}\) & \(S_{\rm i}\) \\ \hline
0.5 & 0.3 \\
0.7 & 0.9 \\
0.4 & 0.2 \\ \(1.6>1.4\) \\ \hline \end{tabular}
\end{table} 
Table 1: Exemplary relationships between \(D_{\rm i}\) and \(S_{\rm i}\)Let \(1\leq u\leq r\) be the number of all \(D_{\rm i}\) such that \(D_{\rm i}\geq S_{\rm i},{\rm i}=1,2,\ldots,r\) By city smartness level, denoted by \(csl\), we understand the following quotient:  
\[csl=\frac{u}{r}\frac{H}{\sum_{i=1}^{r}D_{\rm i}}. \tag{6}\]  
Note that \(csl\in(0,1]\). The city's smartness level could be used to evaluate the degree of smartness the city has achieved. The case \(1\leq u<r\) means that some yet not all city domains are smart, but some are not; in this case \(csl<1\). If however \(csl=1\) (meaning \(u=r\)), then the city is smart. For the sake of narrative lucidity further on, we assumed that \(r=2\), meaning the city is characterized by two agencies (dimensions) only. The values of \(D_{1}\) will be marked on the x axis of the coordinate system, and \(D_{2}\) on the y axis. Figure 1a shows the 2D space within which cities can be found. It is a square with sides of length 1 (this is because they have all been normalized, i.e., \(D_{\rm i}\in(0,1]\), \({\rm i}=1,2,\ldots,r\)--see Formula 1). A city is represented by the couple (\(D_{1}\), \(D_{2}\)). The rectangular area (zone) of smartness is determined by the boundaries of the 2D space and the lines \({\rm x}=S_{1}\) and \({\rm y}=S_{2}\) (colored in grey). Looking at Figure 1a, it is straightforward to note that cities A and B were smart (they lie within the grey area) whereas cities C, D, and E were not smart. City C was smart from the \(D_{2}\) point of view yet was not smart from the \(D_{1}\) perspective, and, therefore, was not smart as such. Having explained the notion of city smartness let us define the notion of sustainability in the context of cities. Intuitively, sustainability means that the set of dimensions of the city \(D=\{D_{1}\), \(D_{2}\),..., \(D_{\rm r}\}\) is balanced, meaning that the values \(D_{1},D_{2},\ldots,D_{\rm r}\) are similar, i.e., \(D_{1}\cong D_{2}\cong\ldots\cong D_{\rm r}\). In the ideal case \(D_{1}=D_{2}=\ldots\ =D_{\rm r}\). For 2D space, such a case is represented by a straight line whose equation is \(y=x\). We dub this line a sustainability line, that is, a line of equilibrium perfectly balanced between the domains. This line can be found in Figure 1a. Thus, sustainable cities should find themselves as close to this line as possible. To describe this observation formally, we constructed a \"corridor\" that forms a neighboring area to the sustainability line, and is constrained by two straight lines lying above and beneath the sustainably line and that keep the same distance from it. Let \(\pi>0\) denote the sustainability distance. The equations for the straight lines determining the corridor are:  
\[y=x+\pi\,\text{and}\ y=x-\pi. \tag{7}\]  
Figure 1: (a) 2D space with the zone of city smartness (in grey) and (b) sustainability corridor. These lines are depicted in Figure 1b. The area \(\mathcal{C}\), such that \(x-\pi\leq C\leq x+\pi\), is called a sustainability corridor. Colloquially, the city is called a sustainable city if it lies within the sustainability corridor. Formally, the city (\(D_{1}\), \(D_{2}\)) is sustainable if, and only if:  
\[D_{2}-\pi\leq D_{1}\leq D_{2}+\pi\text{ or }D_{1}-\pi\leq D_{2}\leq D_{1}+\pi. \tag{8}\]  
It is worth emphasizing that the notion of sustainability and smartness are of different categories. It may well happen that a city is sustainable, i.e., finds itself within the corridor, yet does not satisfy (3) and, therefore, is not smart, which is the case for city F in Figure 1b. In practice it means that the domains of such a city are harmoniously balanced but the value of its synthetic indicator, \(\mathbf{I}\), is low. The notations and definitions introduced above allow us to define the concept of a sustainable smart city.