# Energy Efficient and Reliable Routing Algorithm for Wireless Sensors Networks
## 4 Ds-Eera
### Fusion of Attribute Indexes and Routing Decision  
Based on the data fusion rule of DS evidence theory [24; 33], firstly, the BPA value of transmission energy efficiency ratio (\(m_{1}(A_{\vartheta}^{a})\)) and idleness degree (\(m_{2}(A_{\beta}^{a})\)) of the node \(a\) are fused based on Equation (16) as follows:  
\[m(A^{a})=(m_{1}\oplus m_{2})(A^{a})=\left\{ \begin{array}{cc}0&,A=\Phi\\ \frac{\sum\limits_{A_{\vartheta}\cap A_{\vartheta}\oplus\Phi}\left\{m_{1}(A_{ \vartheta}^{a})m_{2}(A_{\beta}^{a})\right\}}{1-K},A\
eq\Phi,A\subset\Theta\\ \end{array} \right. \tag{17}\]  
where \(K=\sum\limits_{\Phi=A_{\vartheta}\cap A_{\beta}}m_{1}(A_{\vartheta}^{a})m_{2} (A_{\beta}^{a})\), \(\partial=1,2,3\), \(\beta=1,2,3\), \(\Theta=\{A_{1},A_{2},A_{3}\}\), \(A_{1}\) is \"next hop\", \(A_{2}\) is \"non-next hop\", and \(A_{3}\) is \"fuzzy\". In our paper, it should be noted that \(m(A_{1}^{a})\) and \(m(A_{2}^{a})\) are calculated by the Equation (17), but \(m(A_{3}^{a})\) is calculated as follows:  
\[m(A_{3}^{a})=1-m(A_{1}^{a})-m(A_{2}^{a}) \tag{18}\]  
At this point, the BPA value corresponding to each focal element of the first fusion result can be obtained. After obtaining the fusion result \(m(A^{a})\) of the first two BPA value of \(m_{1}(A_{\vartheta}^{a})\) and \(m_{2}(A_{\beta}^{a})\), \(m(A^{a})\) is equivalent to a BPA function of new evidence, which will be fused with the remaining BPA function \(m_{3}(A^{a})\) (energy density factor) by the same DS fusion rule. The value of fusion results \(m(A_{1}^{a})_{final}\), \(m(A_{2}^{a})_{final}\) and \(m(A_{3}^{a})_{final}\) can also be obtained by the above fusion rules. To more clearly describe the routing process of node \(i\) to its forward neighbor nodes, the routing Algorithm 1 is described as follows:  
```
1:Node \(i\), index weight \(w_{b}\), transmission energy efficiency ratio \(P(a)\), idleness degree \(C_{iille}(a)\), energy density factor \(J(a)\), the triangle membership function model \(G\). 2:Output: The next hop \(j\)
3:for\(a=1:m\)\(//m\) is the number of forward neighbor nodes of node \(i\)
4:\(j=1\), \(CN_{next}(i)\leftarrow\Phi\)
5: According to \(G\), traverse \(m\):
6:\(g(a)\gets P(a),g(a)\gets C_{idle}(a),g(a)\gets J(a)\),
7:find\(g_{\max}\leftarrow\max\{g(a)\}\), \(g_{\min}\leftarrow\min\{g(a)\}\),
8: Establish triangular membership function \(G_{t}(g(a))\)
9: Combine \(G_{t}(g(a))\) and corresponding weight to establish BPA function:
10:\(m_{b}(A_{t}^{a})\gets G(g(a)),w_{b}\)
11:Fuse the three indexes by Equations (17) and (18):
12:\(m(A_{t}^{a})\gets m_{1}(A_{t}^{a}),m_{2}(A_{t}^{a})\); \(m(A_{t}^{a})_{final}\gets m_{3}(A_{t}^{a}),m(A_{t}^{a})\)
13:if\(m(A_{1}^{a})>m(A_{2}^{a})\&m(A_{t}^{a})>m(A_{3}^{a})\)\(//\)criterion
14:\(CN_{next}(i)\leftarrow CN_{next}(i)+a\)
15:endif
16:endfor
17:\(j\leftarrow\max(m(A_{1}^{a})_{final}),a\in(CN_{next}(i))\)
18:Return\(j\)
```  
**Algorithm 1** The Next Hop Selection  
The following is a fused example of any two forward neighbor nodes \(a_{1}\) and \(a_{2}\) of node \(i\) according to Algorithm 1. In Table 1, node \(a_{1}\) is obviously judged as \"non-next hop\" under the indexes of transmission energy efficiency ratio and energy density factor, and is judged as \"fuzzy\" under idleness degree. According to DS evidence fusion rule, the fusion result is \(m(A_{2}^{a_{1}})_{final}>m(A_{1}^{a_{1}})_{final}>m(A_{3}^{a_{1}})_{final}\), where \(m(A_{2}^{a_{1}})_{final}\) is the largest, and thus node \(a_{1}\) is judged as \"non-next hop\", not included in candidateneighbor set \(CN_{next}(i)\). For node \(a_{2}\), the fusion result satisfies the criterion and thus belongs to the \(CN_{next}(i)\). By analogy, the fusion results of all forward neighbor nodes of node \(i\) can be calculated, and then the fusion results in \(CN_{next}(i)\) will be sorted. The node with largest \(m(A_{1}^{a})_{final}\) in \(CN_{next}(i)\) will be selected as the next hop of node \(i\). At this point, the next hop of each node and routing path in this round will be determined.