# An Accuracy-Aware Energy-Efficient Multipath Routing Algorithm for WSNs
## 3 Multi-Objective Optimization Model for EMRAR
### Optimization Objects  
1. Reliability object Reliability is evaluated by the transmission success rate (TSR) of the data transmitted from a source node to the sink by multiple paths. It is assumed that bit errors and packet losses are the primary factors impacting the transmission success rate. The TSR can be calculated as follows.  
* For bit error Suppose the data transmission between neighbor nodes is a single hop, and the bit error rate (BER) in the single-hop wireless channel at node \(k\) is \(e_{k}\). Then, the probability of correctly transmitting one bit to the next hop of node \(k\) is \((1-e_{k})\). Then, the probability of correctly transmitting a packet with \(L\) bits on one hop at node \(k\) is \[p_{k}=(1-e_{k})^{L}\] (1) For \(\forall j\), as \(mp_{j}(i,S)\) has \(\left|mp_{j}(i,S)\right|\) hops, then the probability of correctly transmitting a packet from \(i\) to \(S\) on \(mp_{j}(i,S)\) is \[p_{BER_{j}}=\prod_{k=1}^{|mp_{j}(i,\,S)|}p_{k}\] (2) Then, the average probability of correct transmission on BER is \[p_{BER}=\frac{1}{|\mathit{MP}(i,\ S)|}\sum\
olimits_{\begin{subarray}{c}\end{subarray} =1}^{|\mathit{MP}(i,\ S)|}p_{BER_{j}} \tag{3}\]
* For packet loss Suppose the packet loss rate (PLR) for a path \(mp_{j}(i,\ S)\) is \(q_{j}\), then the probability of successfully transmitting a packet from \(i\) to \(S\) on \(mp_{j}(i,\ S)\) is \((1-q_{j})\). Hence, the average probability of successful transmission of PLR is \[p_{PLR}=\frac{1}{|\mathit{MP}(i,\ S)|}\sum\
olimits_{\begin{subarray}{c}\end{subarray} =1}^{|\mathit{MP}(i,\ S)|}(1-q_{j})\] (4)  
For a packet, if it does not arrive at the sink properly, either bit error or packet loss happens. If both happen, it can be treated as packet loss. Then, the average successful transmission probability \(p_{h}\) is  
\[p_{h}=max\{p_{BER},p_{PLR}\} \tag{5}\]  
To meet the demand for reliability, more paths are required. Suppose the paths required to deliver \(M\) coding packets to the sink is \(|\mathit{MP}(i,S)|\) (\(|\mathit{MP}(i,S)|\)\(\leq\)\(M\)). For each packet, the successful transmission probability is \(p_{h}\) and the failure probability is \((1-p_{h})\). The \(TSR\) for transmitting these \(M\) packets on \(|\mathit{MP}(i,S)|\) paths is  
\[TSR=\sum\
olimits_{m=1}^{M}C_{|\mathit{MP}(i,S)|}^{\mathit{m}}(1-p_{h})^{(| \mathit{MP}(i,S)|-m)}p_{h}^{\mathit{m}} \tag{6}\]  
To optimize \(TSR\), we have  
Obj1: Max(\(TSR\))  
2. Energy consumption object In a sensor network, energy consumption is a critical factor of concern. Having a greater number of paths increases reliability, but it also results in higher energy consumption. Suppose the energy consumption for packet transmission from one node to its neighbor is \(E_{p}\). Then, the energy consumption for transmitting a packet on path \(mp_{j}(i,\ S)\) is  
\[E_{j}=E_{p}\sum\
olimits_{\begin{subarray}{c}\end{subarray}=0}^{|mp_{j}(i, \ S)|-1}p_{k} \tag{8}\]  
Therefore, on the multipath transmission with \(|\mathit{MP}(i,S)|\) paths, the total energy consumption is  
\[E_{m}=\sum\
olimits_{\begin{subarray}{c}\end{subarray}=1}^{|\mathit{MP}(i, \ S)|}E_{j}=E_{p}\sum\
olimits_{\begin{subarray}{c}\end{subarray}=1}^{| \mathit{MP}(i,\ S)|}\sum\
olimits_{\begin{subarray}{c}\end{subarray}=0}^{|mp_{ j}(i,\ S)|-1}p_{k} \tag{9}\]  
To optimize \(E_{m}\), we have  
Obj2: Min(\(E_{m}\))