## 4 Interval Improved Fuzzy Partitions Fuzzy C-Means Clustering Algorithm
### Interval Improved Fuzzy Partitions Fuzzy C-Means under City Block Distance  
The new objective function, \(J_{ifpfcmc}\), for the IIFPFCM clustering algorithm is based on a city block distance between the vectors of intervals given in Equation (18) and of which the constraints are given in Equation (19):  
\[ \begin{array}{rcl}J_{ifpfcmc}(\widetilde{G},\ \widetilde{U},\ X)&=&\sum\limits_{i=1}^{c} \sum\limits_{k=1}^{n}\tilde{u}_{ik}^{2}\mathrm{D}_{L2}\left(\overrightarrow{x }_{k},\overset{\widetilde{\delta}}{\widetilde{g}}_{i}\right)-\sum\limits_{k=1 }^{n}\epsilon_{k}^{c}\sum\limits_{l=1}^{c}\left(\widetilde{u}_{ik}-\frac{1}{2 }\right)^{2},\\ &=&\sum\limits_{i=1}^{c}\sum\limits_{k=1}^{n}\tilde{u}_{ik}^{2}\sum\limits_{l =1}^{p}\left(\left|a_{k}^{j}-\widetilde{a}_{i}^{j}\right|+\left|b_{k}^{j}- \widetilde{\beta}_{i}^{j}\right|\right)-\sum\limits_{k=1}^{n}\widetilde{ \epsilon}_{k}\sum\limits_{i=1}^{c}\left(\widetilde{u}_{ik}-\frac{1}{2}\right)^ {2},\end{array}  \tag{18}\]  
\[\text{subject to }\sum\limits_{i=1}^{c}\widetilde{u}_{ik}=1. \tag{19}\]  
In Equation (18), \(\widetilde{u}_{ik}\) is the fuzzy membership degree of pattern \(k\) in the \(i\)th cluster, and \(\widetilde{U}=[\widetilde{u}_{ik}]\) is a \(c\times n\) matrix; the \(\widetilde{\epsilon}_{k}\) is similar to the IFPFCM defined in Equation (20):  
\[\widetilde{\epsilon}_{k}=\min\Biggl{\{}\sum\limits_{j=1}^{p}\left(\left|a_{k} ^{j}-\widetilde{a}_{i}^{j}\right|+\left|b_{k}^{j}-\widetilde{\beta}_{i}^{j} \right|\right)-\eta|i\in\{1,\ 2,\ \ldots,\ c\}\Biggr{\}},\ \eta\in(0,\ \infty). \tag{20}\]  
Using the Lagrange multipliers method for the objective function \(J_{ifpfcmc}\), the nonlinear unconstrained optimization can be obtained as:  
\[L_{ifpfcmc}=\sum\limits_{i=1}^{c}\sum\limits_{k=1}^{n}\widetilde{u}_{ik}^{2} \sum\limits_{j=1}^{p}\left(\left|a_{k}^{j}-\widetilde{\alpha}_{i}^{j}\right|+ \left|b_{k}^{j}-\widetilde{\beta}_{i}^{j}\right|\right)-\sum\limits_{k=1}^{n} \widetilde{\epsilon}_{k}\sum\limits_{i=1}^{c}\left(\widetilde{u}_{ik}-\frac{1} {2}\right)^{2}-\sum\limits_{k=1}^{n}\lambda_{k}\Biggl{(}\sum\limits_{i=1}^{c} \widetilde{u}_{ik}-1\Biggr{)}. \tag{21}\]  
The \(\widetilde{u}_{ik}\) was considered for the minimum problem; to obtain an updated equation of the fuzzy membership degree, \(u_{ik}\) can be solved as followed:  
\[\frac{\partial L_{ifpfcmc}}{\partial\widetilde{u}_{ik}}=0\Leftrightarrow 2 \widetilde{u}_{ik}\sum\limits_{j=1}^{p}\left(\left|a_{k}^{j}-\widetilde{\alpha }_{i}^{j}\right|+\left|b_{k}^{j}-\widetilde{\beta}_{i}^{j}\right|\right)-2 \widetilde{\epsilon}_{k}\left(\widetilde{u}_{ik}-\frac{1}{2}\right)-\lambda_{k }=0, \tag{22}\]  
for \(i=1,\ \ldots,\ c\) and \(k=1,\ \ldots,\ n\). Thus, Equation (22) can be rewritten as:  
\[\widetilde{u}_{ik}=\frac{\lambda_{k}-\widetilde{\epsilon}_{k}}{2\left(\sum \limits_{j=1}^{p}\left(\left|a_{k}^{j}-\widetilde{\alpha}_{i}^{j}\right|+ \left|b_{k}^{j}-\widetilde{\beta}_{i}^{j}\right|\right)-\widetilde{\epsilon}_{ k}\right)}, \tag{23}\]  
for \(i=1,\ \ldots,\ c\) and \(k=1,\ \ldots,\ n\). Substituting Equation (23) into Equation (19) is obtained as:  
\[\sum\limits_{s=1}^{c}\frac{\lambda_{k}-\widetilde{\epsilon}_{k}}{2\left(\sum \limits_{j=1}^{p}\left(\left|a_{k}^{j}-\widetilde{\alpha}_{s}^{j}\right|+ \left|b_{k}^{j}-\widetilde{\beta}_{s}^{j}\right|\right)-\widetilde{\epsilon}_{ k}\right)}=1. \tag{24}\]According to Equations (23) and (24), the updated equation for the fuzzy membership degree \(\widetilde{u}_{ik}\) is:  
\[\widetilde{u}_{ik}=\frac{1}{\sum\limits_{s=1}^{c}\left(\frac{\sum\limits_{j=1}^{ p}\left(\left|\left|\delta_{k}^{j}-\widetilde{s}_{i}^{j}\right|+\left|\left| \psi_{k}^{j}-\widetilde{\beta}_{i}^{j}\right|\right)-\widetilde{\alpha}_{k}}{ \sum\limits_{j=1}^{p}\left(\left|\left|\delta_{k}^{j}-\widetilde{s}_{i}^{j} \right|+\left|\psi_{k}^{j}-\widetilde{\beta}_{i}^{j}\right|\right)-\widetilde{ \alpha}_{k}}\right)}\right., \tag{25}\]  
for \(i=1,\ \ldots,\ c\) and \(k=1,\ \ldots,\ n\).