## 4 SD-EAR in Detail
### Optimum Route Selection
#### 4.2.2 Inter-Zone Route Discovery with Example
If any one of the f-en\({}_{1}\)(R) and f-en\({}_{2}\)(R) is 0, then the effect of the other one will be completely eliminated, which is undesirable. **trim(R) or the transmission energy impact of route R**  
Transmission energy impact depends on the minimum received power of the nodes in route R and the maximum distance between consecutive nodes. With the increase in this distance and the minimum received power of the successors (by successor we mean all of the nodes except the source) in R, the transmission cost increases, while trim(R) decreases. \[\text{trim}(\text{R})=\sqrt{(\text{f-recv}_{1}(\text{R})\ \times\ \text{f-recv}_{2}(\text{R}))} \tag{26}\]  
\[\text{f-recv}_{1}(\text{R})=(1-\sum\text{mrpw}(\text{succ}(n_{k},\text{R}))/ (\alpha(\text{R})\times(\text{max mrpw}\ (n_{\text{v}})+1))\]  
\[\text{n}_{k}\in\text{R}\]  
\[\text{f-recv}_{2}(\text{R})=\max\{1,\text{dist}(n_{k},\text{succ}(n_{k},\text{ R}))\}\]  
\[\text{n}_{k}\in\text{R}\]0  
succ(n\({}_{k}\), R) is the successor node of n\({}_{k}\) in R. mrpw(n\({}_{v}\)) is the minimum received power of node n\({}_{v}\). dist(n\({}_{a}\), n\({}_{b}\)) specifies the Cartesian distance between nodes n\({}_{a}\) and n\({}_{b}\). From the formulation in Equation (26), we can see that trim lies between 0 and 1. High values of it contribute to improve the performance of R.  
**neff(R) or node efficiency in R**  
The node efficiency in a path will be high, as most of the nodes are awake, and alternatives are available to sleeping nodes (if any). A node \(\rm n_{u}\) will be an alternative to another node \(\rm n_{v}\) w.r.t. route provided the following conditions are satisfied:  
\[\text{n}_{k}\in\text{R}\]1  
\[\text{n}_{k}\in\text{R}\]2  
\[\text{n}_{k}\in\text{R}\]3  
Condition (27a) specifies that \(\rm n_{u}\) should not belong to any live communication route passing through \(\rm n_{v}\). Condition (27b) says that \(\rm n_{v}\) must have all of the neighbors of \(\rm n_{u}\) as its neighbors. The condition (27c) indicates that the additional forwarding load from neighbors of \(\rm n_{u}\) should not reduce the lifetime of \(\rm n_{v}\) below the lifetime of \(\rm n_{u}\). If \(\rm n_{u}\) is an alternative of \(\rm n_{v}\), then \(\rm n_{u}\in alt(v)\) where alt(v) is the set of all of the alternatives of \(\rm n_{v}\). The mathematical formulation of neff appears in Equation (28):  
\[\text{n}_{k}\in\text{R}\]4  
\[\text{n}_{k}\in\text{R}\]5  
\[\text{n}_{k}\in\text{R}\]6  
\[\text{n}_{k}\in\text{R}\]7  
\[\text{n}_{k}\in\text{R}\]8  
\[\text{n}_{k}\in\text{R}\]9  
From Equation (28), it is evident that the efficiency of a route increases with the number of awake nodes and the number of alternatives to sleepy nodes. neff lies between 0 and 1. High values of it indicate the efficiency of the associated route. **FUZZY RULE BASES OF FUZZ-OPT-ROUTE**  
All of the input parameters of the FUZZ-OPT-ROUTE are divided into four equal ranges between 0 and 1. The ranges and corresponding fuzzy premise variables are as follows:  
(i) 0-0.25 (LOW or 'L' is short)  
(ii) 0.25-0.50 (SATISFACTORY or 'S' in short)  
(iii) 0.50-0.75 (HIGH or 'H' in short)  
(iv) 0.75-1.00 (VERY HIGH or 'VH' in short)  
The rule bases of this fuzzy controller appear in Tables 1 and 2. Table 1 combines reft and trim, producing temporary output temp1. Both of these parameters are assigned equal weight, since both are equally important in determining the efficiency of the route. If a node has high residual energy but suffers from huge energy consumption due to the long distance from its successor in a communication path, then the energy efficiency of a path greatly reduces. Therefore, temp1 = min (reft, trim) in \[\text{k}\
eq\text{start}(\text{R})\ \text{and}\]0 Examples are: reft = H, trim = S, temp1 = S; reft = S, trim = L, temp1 = L; and reft = VH, trim = S, temp1 = S. The composition of temp1 and neff is presented in \[\text{k}\
eq\text{start}(\text{R})\ \text{and}\]1 Then eff of a route increases with an increased number of awake nodes and alternatives to sleepy nodes, if any.