# An Energy Efficient Routing Approach for Cloud-Assisted Green Industrial IoT Networks
## 5 Proposed Energy Efficient and Reliable Routing Approach
### Primitive Requirements for Routing Metric  
The use of the RAROF is to optimize the RPL performance for emergency monitoring IIoT network, so the applied metric needs to comply with the properties of consistency, optimality, and loop-freeness as defined in [48]. Let's assume, \(Rp\) is a path from the source to the destination node, and \(v^{\prime}_{j}\) is the on-path node, then \(\varphi(Rp,v^{\prime}_{j})\) signifies the next-hop of \(v^{\prime}_{j}\) on \(Rp\). In this case, if \(v^{\prime}_{j}\) is the destination node of \(Rp\), then \(\varphi(Rp,v^{\prime}_{j})\ =\ NULL\). Therefore, \(\varphi(Rp,v^{\prime}_{j})\) represents the next-hop on the path to the destination in a multi-hop routing in which \(v^{\prime}_{j}\) stores routing information in its table. Each node updates the information when the trickle timer is reset. As explained in [16], the DAG metric container object of DIO message carries the routing metrics and other network information. In the proposed RAROF, the route selection mechanism is represented by a mapping function \(\Gamma((Rp_{i\to j})\in\mathbb{R})\), which determines the route by calculating the different metric values (i.e., remaining energy, buffer occupancy, ETX, hop count, delay, etc.) of a path between the source node \(i\) and the destination node \(j\). As a result, each path between nodes \(i\) and \(j\) has a cost (path weight) that can be ordered, leading to the best path selection, and packets can be routed along the lightest path.  
A routing protocol is consistent if data packets from any source node \(v_{i}\in V\) to any destination node \(v_{j}\in V\) are delivered through any on-path node \(v^{\prime}_{j}\) of \(\Gamma(Rp_{i\to j})\), then other nodes on \(\Gamma(Rp_{i\to j})\) must take the same decision; i.e., \(v^{\prime}_{j}\) forwards packets from \(v_{i}\) to \(v^{\prime}_{j}\) to \(\varphi(\Gamma(Rp_{i\to j}),v^{\prime}_{j})\). According to [49], the routing protocol is known to be optimal if it forwards the data packets through the lightest route (minimum latency, overhead, and energy consumption, etc.) between every pair of nodes in the network. As described in [48; 50], the isotonicity and monotonicity characteristics of the routing metric are pertinent to ensure whether a routing protocol can satisfy the consistency, optimality, and loop-free requirements during path construction. Therefore, in this paper, we only consider the isotonicity and monotonicity properties of the routing metrics as follows.  
Suppose \(\leq\) is used to characterize an order relation, and \(\oplus\) is the operator used to concatenate any two routes. The isotonicity of the routing metric states that the order relation among the weights of any two routes is preserved when both routes are prefixed or suffixed by a third common route. It can be expressed mathematically, as follows.  
\[\Gamma(Rp_{v_{1}\to v_{j}})\leq\Gamma(Rp_{v_{2}\to v_{1}})\Rightarrow \left\{\begin{array}{l}\Gamma(Rp_{v_{1}\to v_{2}}\oplus Rp_{v_{m}\to v_{n}}) \leq\Gamma(Rp_{v_{k}\to v_{l}}\oplus Rp_{v_{m}\to v_{n}})\\ \Gamma(Rp_{v_{0}\to v_{2}}\oplus Rp_{v_{1}\to v_{2}})\leq\Gamma(Rp_{v_{0}\to v_{ 2}}\oplus Rp_{v_{k}\to v_{l}})\end{array}\right.\]  
\[\forall Rp_{v_{1}\to v_{2}},Rp_{v_{2}\to v_{1}},Rp_{v_{m}\to v_{n}},Rp_{v_{0} \to v_{p}}\in\mathbb{R}\]On the other hand, the routing metric is monotonic if the path cost or weight does not improve when prefixed or suffixed by another path. Mathematically, it can be expressed as follow.  
\[\left\{\begin{array}{l}\Gamma(Rp_{l\to j})\leq\Gamma(Rp_{l\to j}\oplus Rp_{j\to k})\\ \Gamma(Rp_{l\to j})\leq\Gamma(Rp_{k\to l}\oplus Rp_{l\to j})\end{array}\right., \forall Rp_{i\to j},Rp_{j\to k},Rp_{k\to i}\in\mathbb{R}\]