# Multiagency Modeling of Transformation Strategies Towards Sustainable Smart Cities
## 3 The Multiagency Sustainable Smart City Model and Transformation Strategies
### Model Description  
Our city model is defined in the following way. The city is characterized by a finite set, \(D=\{D_{1},D_{2},\)\(\ldots,D_{r}\}\) composed of r dimensions (city agencies), where each dimension \(D_{i}\) is a finite set of its features \(\Gamma^{i}_{D_{i}^{i}}\), i.e., \(D_{i}=\left\{\Gamma^{1}_{D_{i}^{i}},\Gamma^{2}_{D_{i}^{i}},\ldots,\Gamma^{n} _{D_{i}^{i}}\right\}\) such that \(\left\{\Gamma^{1}_{D_{i}^{i}}\cup\Gamma^{2}_{D_{i}^{i}}\cup\ldots\cup\Gamma^ {ni}_{D_{i}^{i}}\right\}\
eq\varnothing,\text{fori}=1,2,\ldots,r;j=1,2,\ldots,n\). To each feature \(\Gamma^{j}_{D_{i}^{i}}\) is assigned its value \(\alpha^{j}_{D_{i}^{i}}\
eq 0\) (a positive real number) and it is weighted by the value \(w^{j}_{D_{i}^{i}}\) for \(i=1,2,\ldots,r;j=1,2,\ldots,n\). Let \(D_{i}\) denote the value to be assigned to \(D_{i}\). Formally, the following formulae determine the model:  
\[D_{1}=\frac{w^{1}_{D1}\alpha^{1}_{D1}+w^{2}_{D1}\alpha^{2}_{D1}+\ldots+w^{n}_{ D1}\alpha^{n}_{D1}}{a^{1}_{D1}+a^{2}_{D1}+\ldots+a^{n}_{D1}},\]  
where \(0\leq w^{j}_{D1}\leq 1,j=1,2,\ldots,n\). \[D_{2} =\frac{w_{D2}^{1}a_{D2}^{1}+w_{D2}^{2}a_{D2}^{2}+\ldots+w_{D2}^{m}a_{D2} ^{m}}{a_{D2}^{1}+a_{D2}^{2}+\ldots+a_{D2}^{m}}, \tag{1}\]  
where \(0\leq w_{D2}^{j}\leq 1,j=1,2,\ldots,m\). \[D_{\rm r} =\frac{w_{D\rm r}^{1}a_{D\rm r}^{1}+w_{D\rm r}^{2}a_{D\rm r}^{2}+ \ldots+w_{D\rm r}^{n}a_{D\rm r}^{p}}{a_{D\rm r}^{1}+a_{D\rm r}^{2}+\ldots+a_{D \rm r}^{p}},\]  
where \(0\leq w_{D\rm r}^{j}\leq 1,j=1,2,\ldots,p\). Note that as a result of normalization, \(D_{\rm i}\in(0,1]\) for any \({\rm i}=1,2,\ldots,r\). The formula to calculate the synthetic indicator, \(I\in(0,1]\), that provides a general description of the city is the following:  
\[I =\frac{g_{1}D_{1}+g_{2}D_{2}+\ldots+g_{\rm r}D_{\rm r}}{D_{1}+D_{2}+ \ldots+D_{r}}, \tag{2}\]  
where \(0\leq g_{\rm i}\leq 1\) is the weight associated with \(D_{\rm i}\), \({\rm i}=1,2,\ldots,r\). The above model does not take into account the possible relationships occurring between the dimensions \(D=\{D_{1},\,D_{2},\,\ldots,\,D_{\rm r}\}\). In order to enhance the model, the relationships can be modeled by means of Cartesian products over \(D\). Let \(\mathbb{D}_{\rm i}\) denote an i-ary Cartesian product over \(D\), for example,\(\mathbb{D}_{3}=D\times D\times D\). So, the set of all relationships between the dimensions is defined as follows: \(\cup_{i=2}^{r}\mathbb{D}_{\rm i}\). Obviously not all possible relationships occur, therefore, what actually takes place is a subset of \(\cup_{i=2}^{r}\mathbb{D}_{\rm i}\). This approach for modelling the relationships can, in a similar way, go deeper down to the level of features \(\Gamma_{D_{\rm i}}^{j}\); this exercise is left to the reader. For each dimension, \(D_{\rm i}\), we assigned its smartness threshold, \(S_{\rm i}\) (\({\rm i}=1,2,\,\ldots,\,{\rm r}\)). Now, we were able to formally define what is meant by the term smart city. The city whose dimensions take on the values \(D_{1},\,D_{2},\,\ldots,\,D_{\rm r}\) is smart if, and only if,  
\[S_{\rm i}\leq D_{\rm i}\leq 1, \tag{3}\]  
for any \({\rm i}\in\{1,2,\,\ldots,\,r\}\). Note that in the case where:  
\[\sum_{{\rm i}=1}^{r}D_{\rm i}\geq\sum_{{\rm i}=1}^{r}S_{\rm i}, \tag{4}\]  
does not guarantee that the city is smart. A simple example given in Table 1 proves this statement. The city described in the above table is not smart because \(D_{2}<S_{2}\) (\(0.7<0.9\)) despite fulfilling (4). It might be convenient to introduce the notion of local smartness. The domain \(D_{\rm i},\,i\in\{1,2,\ldots,r\}\), is locally smart if \(D_{\rm i}\geq S_{\rm i}\), where \(S_{i}\) is the smartness threshold assigned to this domain. Let us consider all the domains, \(D_{\rm i},{\rm i}=1,2,\ldots,r\), whose values, \(D_{\rm i}\), are greater or equal to \(S_{\rm i}\), and calculate their sum:  
\[H=\sum_{{\rm i}=1}^{r}D_{\rm i},\mbox{ for }\,D_{\rm i}\geq S_{\rm i},{\rm i}=1,2, \ldots,r. \tag{5}\]