## 4 The Proposed Scheme HFECS
### CH Election Procedure  
Based on the sensory data being generated by sensor nodes, we make a decision about CH role in the cluster using \(TF_{V_{i}V_{j}}\), \(R_{E}\), \(LQR\), \(D_{toBS}\) and \(N_{support}\). But before that, we have to construct the hesitant fuzzy entropy matrix and determine the corresponding weight coefficents of each attribute. Subsequently, the attribute values are synthesized to determine the threshold attribute values. The threshold attribute values are then compared with original attribute values of each sensor node. Based on their comparison, the BS selects the sensor nodes with original attribute values greater than threshold attribute values and add those sensor nodes in CH election set. The entropy measures for hesitant fuzzy set were already discussed by [PERSON] et al. in [40] and [PERSON] et al. in [41]. Due to the fact that hesitancy is a common problem in decision making problems, we need to develop some entropy or cross entropy measures for hesitant fuzzy sets. [PERSON] et al. in [19] discussed that if X is a fixed set and the hesitant fuzzy set \(\alpha\) when applied on X returns a subset in [0,1]. Let \(l_{\alpha(x)}\) be the number of values in \(\alpha(x)\) where \(\alpha(x)_{o_{i}}\) be the \(i\)th smallest value in \(\alpha(x)\) in which \(i=\left\{1,2,\ldots l_{\alpha(x)}\right\}\). In order to find the cross entropy of two hesitant fuzzy sets, i.e., \(\alpha\) and \(\beta\), we assume that both of them have same length, or, if there is only one value in \(\alpha\), we extend it by repeating that value until it reaches the length of \(\beta\). Moreover, if there is only one hesitant fuzzy set, then we use \(\alpha\) and \(\alpha^{C}\)(complement of \(\alpha\)) in the cross-entropy and calculate the hesitant fuzzy entropy in Equations (8) and (9) as  
\[E_{A}(\alpha)\ =\ 1\ -\ C_{A}\Big{(}\alpha,\ \alpha^{C}\Big{)} \tag{8}\]where \(C_{A}\big{(}\alpha,\,\alpha^{C}\big{)}=\frac{2}{l_{\alpha}T}\sum\limits_{i=1}^{l_{ \alpha}}\left\{\frac{\big{(}1+q\alpha_{\sigma(i)}\big{)}\ln\big{(}1+q\alpha_{ \sigma(i)}\big{)}+\big{(}1+q\big{(}1-\alpha_{\sigma(l_{\alpha}-i+1)}\big{)} \big{)}\ln\big{(}1+q\big{(}1-\alpha_{\sigma(l_{\alpha}-i+1)}\big{)}\big{)}}{2}- \frac{2+q\alpha_{\sigma(i)}\,\,q\big{(}1-\alpha_{\sigma(l_{\alpha}-i+1)}\big{)} }{2}\ln\frac{2+q\alpha_{\sigma(i)}\,\,q\big{(}1-\alpha_{\sigma(l_{\alpha}-i+1) }\big{)}}{2}\right)q>0\)  
\[ \begin{array}{rcl}E_{A}(\alpha)&=&1\,-\,\frac{2}{l_{\alpha}T}\sum\limits_{i= 1}^{l_{\alpha}}\left(\frac{\big{(}1+q\alpha_{\sigma(i)}\big{)}\ln\big{(}1+q \alpha_{\sigma(i)}\big{)}+\big{(}1+q\big{(}1-\alpha_{\sigma(l_{\alpha}-i+1)} \big{)}\big{)}\ln\big{(}1+q\big{(}1-\alpha_{\sigma(l_{\alpha}-i+1)}\big{)} \big{)}}{2}\right.\\ &&\qquad\qquad\qquad\qquad\qquad\qquad\left.-\,\frac{2+q\alpha_{\sigma(i)}\, \,q\big{(}1-\alpha_{\sigma(l_{\alpha}-i+1)}\big{)}}{2}\ln\frac{2+q\alpha_{ \sigma(i)}\,\,q\big{(}1-\alpha_{\sigma(l_{\alpha}-i+1)}\big{)}}{2}\right)q>0 \end{array}  \tag{9}\]  
where \(T=(1+q)\ln(1+q)-(2+q)(\ln(2+q)-\ln(2)),\,\,q>0\). Then, we calculate the entropy weight coefficients for all sensor nodes using Equation (10) after generating entropy matrix using Equation (9). \[w_{i}=\,\,\frac{1-E_{i}}{r-\sum\limits_{i=1}^{r}E_{i}} \tag{10}\]  
where \(0\leq w_{i}\leq 1\), \(\sum\limits_{i=1}^{r}w_{i}=1\) and \(E_{i}=\frac{1}{r}\,\,\sum\limits_{i=1}^{r}E\big{(}\alpha_{ji}\big{)}\). In the next step, we find the average data value of attributes in measured value matrix, i.e., \(D_{i}=\frac{1}{l_{\alpha}}\,\,\sum\limits_{i=1}^{l_{\alpha}}\,\alpha_{\sigma( i)}\) and then synthesize the value of each attribute to find the threshold attribute value using Equation (11) as:  
\[D_{Fi}=\frac{1}{r}\,\,\sum\limits_{i=1}^{r}w_{i}D_{i} \tag{11}\]  
The decision about placing sensor nodes in CH election set depends on the comparison between the original attribute values of sensor nodes in a cluster with that of threshold attribute values.