## 4 Proposed Scheme
### New Node-Association
Weights Assignment After normalization, weights are assigned to each criterion as shown in Equation (7). The \(w_{j}^{*}\) in Equation (7) is a weight value allocated to each resource criterion, \(j\). The assignments of weights are application-specific (e.g., weights may vary from application to application). \[X^{\prime}=\left( \begin{array}{cccccc}w_{1}^{*}\times{X^{*}}_{(1,1)}&w_{2}^{* }\times{X^{*}}_{(1,2)}&w_{3}^{*}\times{X^{*}}_{(1,3)}&.&.&w_{m}^{*}\times{X^{* }}_{(1,n)}\\ w_{1}^{*}\times{X^{*}}_{(2,1)}&w_{2}^{*}\times{X^{*}}_{(2,2)}&w_{3}^{*}\times{X^ {*}}_{(2,3)}&.&.&w_{m}^{*}\times{X^{*}}_{(2,n)}\\.&.&.&.&.&.\\.&.&.&.&.&.\\.&.&.&.&.&.\\ w_{1}^{*}\times{X^{*}}_{(m,1)}&w_{2}^{*}\times{X^{*}}_{(m,2)}&w_{3}^{*}\times{X^ {*}}_{(m,3)}&.&.&w_{m}^{*}\times{X^{*}}_{(m,n)}\end{array} \right)\] (7)
4. Ideal Positive Solutions (\(IPS^{+}\)) and Ideal Negative Solutions (\(INS^{-}\)) _Ideal Positive Solution (\(IPS^{+}\)):_ The resource criteria where high attribute values such as residual energy, computational capability, and storage capacity are desired are named as \(IPS^{+}\). The residual energy of the node is very important, as the overall lifetime of the node depends on its residual energy. Therefore, the highest value of energy is taken as \(IPS^{+}\). Similarly, the high storage capacity is also an important criterion, since providing the large storage to a CH prevents congestion and packet drop on a CH. Likewise, the high computational capability reduces the processing delays. All these aforementioned criteria optimize the packet drop rate and enhance the performance of the network. \(IPS^{+}\) is computed using Equation (8) and is defined as follows: \[IPS^{+}=\max\{(X^{{}^{\prime}}_{1,j}),\ (X^{{}^{\prime}}_{2,j})..(X^{{}^{ \prime}}_{m,j})\},\forall\,j\in\,n\] (8) Ideal Negative Solution \(INS^{-}\): The resource criteria where the low attribute values such as traffic load on CH, distance from CH to the sink node, and distance from a child node to the CH are desired considered as \(INS^{-}\). For instance, if there is a high traffic load on CH, the CH cannot accommodate more data packets from its child nodes due to storage limitations. Similarly, If the distance between the sink node and CH is high, the CH consumes high amount of energy in data transmission towards the sink node. Likewise, if the distance between the child node and a CH is high, the child node has to consume high mount of energy to transmit data. The high values of the aforementioned resource criteria(s) are not beneficial; therefore, these criteria are considered as \(INS^{-}\). \(INS^{-}\) is computed using Equation (9) and is defined as follows. \[INS^{-}=\min\{(X_{1,j}^{{}^{\prime}}),\ (X_{2,j}^{{}^{\prime}})..(X_{m,j}^{{}^{ \prime}})\},\forall\;j\in\;n \tag{9}\]
5. Difference of each CH from \(IPS^{+}\) and \(INS^{-}\) For each criterion, the difference of each resource criteria from the \(IPS^{+}\) and \(INS^{-}\) are calculated using Equations (10) and (11), respectively and are defined as follows. \[X_{i}^{+}=\sqrt{\sum\limits_{j=1}^{n}(X_{(i,j)}^{{}^{\prime}}-\ IPS^{+})^{2}}\; \textit{where}\;i=\;1...m\] (10) \[X_{i}^{-}=\sqrt{\sum\limits_{j=1}^{n}(X_{(i,j)}^{{}^{\prime}}-\ INS^{-})^{2}}\; \textit{where}\;i=\;1...m\] (11) where, \(X_{i}^{+}\) and \(X_{i}^{-}\) represent the difference of CH candidate from \(IPS^{+}\) and \(INS^{-}\) respectively. 6. Ranking Index for final decision Equations (10) and (11) compute the difference of each CH node from \(IPS^{+}\) and \(INS^{-}\) respectively. By doing so, the child node obtains a deviated value of all CHs from \(IPS^{+}\) and \(INS^{-}\). Since the CHs are heterogeneous in terms available resource, it is highly likely that the deviated value of each CH varies from each other. Based on these deviated values, the child node computes the rank index (\(R_{i}\)) of each CH node by employing Equation (12). The \(R_{i}\) computed from Equation (12) guarantees that the CH with the best available resources is assigned a highest rank as compare to the other potential CHs.