# An Energy Efficient Routing Approach for Cloud-Assisted Green Industrial IoT Networks
## 5 Proposed Energy Efficient and Reliable Routing Approach
### The Routing Metrics and RAROF Design
In this entire process, a node \(v_{j}\) can be a candidate parent node of node \(v_{i}\), only if it satisfies the following conditions. \[p\in NB(v_{i});\,v_{j}\in NB(p);\,r(p)\prec r(v_{j}) \tag{10}\]  
where \(r\) denotes the rank of a node. Unlike default RPL implementation [16], in this paper, the preferred parent selection and the rank calculation is detached. Thus, RAROF constructs the DODAG tree structure based on the node's rank, for which the ETX metric is used, and the rank of a node \(r(v_{i})\) is computed as in Equation (11). \[r(v_{i})\ =\ \min_{j\in NB(i)}(lq(v_{i},v_{j})+r(Pp(v_{i}))+\gamma \tag{11}\]  
Here, \(r(Pp(v_{i}))\) is the rank value of the preferred parent of the node \(v_{i}\), and \(\gamma\) is the constant integer used to confirm the loop-freeness in the network and satisfying the Equation (10). It is defined to ensure the rank increment of a node that satisfies the rank policy of the DODAG-tree structure specified in [16]. Finally, the proposed RAROF selects the next-hop node based on the minimum function and determines the data forwarding path as follows. \[ \begin{array}{l}\min\Gamma NVI(v_{i})+R_{av}\\ s.t.\,R_{av}\ =\ 1\end{array}  \tag{12}\]  
Furthermore, the proposed metric also meets the primitive requirements such as isotonicity and monotonicity, the proof is as follows. Let us assume that there exist \(n\) routing paths \(Rp_{n}\in\mathbb{R}\), where \(n=3\). These three routing paths are represented as \(Rp_{1}\ =\ \mathbb{R}(v_{i},v_{i})\ =\ (v_{i},v_{i+1},v_{i+2},v_{i+3},\ldots,v_{j}),Rp_{2} =\mathbb{R}(v^{\prime}_{i},v^{\prime}_{i+1},v^{\prime}_{i+2},v^{\prime}_{i+3},\ldots,v^{\prime}_{j})\), and \(Rp_{3}\ =\ \mathbb{R}(v^{\prime\prime}_{i},v^{\prime\prime}_{j})\ =\ (v^{\prime\prime}_{i},v^{\prime\prime}_{i+1},v^{\prime \prime}_{i+2},v^{\prime\prime}_{i+3},\ldots,v^{\prime}_{j})\). Next, the metric values of these paths are as follows. \[\Gamma(NVI(Rp_{1}))\ =\ \Gamma(NVI(v_{i},v_{i+1})\oplus NVI(v_{i},v_{i+1}) \oplus\ldots\oplus NVI(v_{j-1},v_{j}))\ =\ \sum\limits_{x\ =\ 1}^{n}\Gamma NVI(v_{x})\geq 0\] \[\Gamma(NVI(Rp_{2}))\ =\ \Gamma(NVI(v^{\prime}_{i},v^{\prime}_{i+1}) \oplus NVI(v^{\prime}_{i},v^{\prime}_{i+1})\oplus\ldots\oplus NVI(v_{j-1},v_{ j}))\ =\ \sum\limits_{x\ =\ 1}^{n}\Gamma NVI(v^{\prime}_{x})\geq 0\] \[\Gamma(NVI(Rp_{3}))\ =\ \Gamma(NVI(v^{\prime\prime}_{i},v^{\prime\prime}_{i+1}) \oplus NVI(v^{\prime\prime}_{i},v^{\prime\prime}_{i+1})\oplus\ldots\oplus NVI(v ^{\prime\prime}_{j-1},v^{\prime\prime}_{j}))\ =\ \sum\limits_{x\ =\ 1}^{n}\Gamma NVI(v^{\prime\prime}_{x})\geq 0\]  
Here, \(x\) is the traversed node in \(Rp_{n}\in\mathbb{R}\). The metric value of each routing path is always \(\geq 0\). If the path function \(\Gamma NVI(Rp_{1})\leq\Gamma NVI(Rp_{3})\), then \(\Gamma NVI(Rp_{1})+\Gamma NVI(Rp_{2})\leq\Gamma NVI(Rp_{3})+\Gamma NVI(Rp_{2})\) can be easily proved, regardless of the prefixed or suffixed by third common path such as \(Rp_{2}\). Therefore, it is proved that the proposed metric satisfies the isotonicity requirement. On the other hand, the metric value of each routing path is always \(\geq 0\), so, \(\Gamma NVI(Rp_{1})\leq\Gamma NVI(Rp_{1})+\Gamma NVI(Rp_{2})\) can be easily proved regardless of the prefixed or suffixed by a third common path such as \(Rp_{2}\). In this way, it also proves that the proposed metric meets the monotonic requirement.