## 2 System Model and Problem Formulation
We define as a binary assignment indicator:  
\[\varepsilon_{k}^{l}=\left\{ \begin{array}{ll}1&\mbox{if the $l$th relay is assigned to the $k$th user}\\ 0&\mbox{otherwise}\end{array} \right.\]  
The channel capacity of the \(k\)th user for AF relaying with the binary assignment indicator is:  
\[C_{k}^{\mathsf{g}}=\frac{1}{2}\log\left[1+\frac{p_{k}^{s}}{N}\left(|h_{k}^{s} |^{2}+\Omega_{k}^{\mathsf{g}}\right)\right] \tag{2}\]  
where \(\Omega_{k}^{\mathsf{g}}=\left(\frac{\left(\sum_{l=1}^{L}\varepsilon_{k}^{l}|h_ {l}^{s}h_{l}^{l}|\beta_{l}\sqrt{p}\right)^{2}}{1+\sum_{l=1}^{L}\varepsilon_{k} ^{s}\left(|h_{k}^{s}|\beta_{l}\sqrt{p}\right)^{2}}\right)\). Our first objective is to maximize the sum-rate capacity, \(\sum_{k=1}^{K}C_{k}\). To normalize the first objective between 0 and 1, we will divide the sum-rate capacity with its upper bound, \(\sum_{k=1}^{K}C_{k}^{max}\). **Lemma 1**. : _The decision variable-free upper bound of Equation (1) is \(C_{k}^{max}=\frac{1}{2}\log\left[1+\frac{P_{s}}{N}\left(|h_{k}^{s}|^{2}+\sum_ {l}|h_{l}^{s}|^{2}\right)\right]\)._  
Proof. : It is easy to see that \(C_{k}\) in Equation (1) is an increasing function of the source power. We can set the source power to its maximum transmission power, \(P_{s}\). We will get an upper bound as:  
\[ \begin{split} C_{k}&=\frac{1}{2}\log\left[1+\frac{p_ {k}^{s}}{N}\left(|h_{k}^{s}|^{2}+\frac{\left(\sum_{l}|h_{l}^{s}h_{k}^{l}|\beta_ {l}\sqrt{p_{l}}\right)^{2}}{1+\sum_{l}\left(|h_{k}^{l}|\beta_{l}\sqrt{p_{l}} \right)^{2}}\right)\right]\\ &<\frac{1}{2}\log\left[1+\frac{P_{s}}{N}\left(|h_{k}^{s}|^{2}+ \frac{\left(\sum_{l}|h_{l}^{s}h_{k}^{l}|\beta_{l}\sqrt{p}\right)^{2}}{\sum_{l} \left(|h_{k}^{l}|\beta_{l}\sqrt{p}\right)^{2}}\right)\right]\end{split}  \tag{3}\]  
Applying Cauchy-Schwartz inequality, we will get:  
\[ \begin{split}&\leq\frac{1}{2}\log\left[1+\frac{P_{s}}{N}\left(|h_{k }^{s}|^{2}+\frac{\sum_{l}|h_{l}^{s}|^{2}\sum_{l}\left(|h_{k}^{l}|\beta_{l}\sqrt{ p}\right)^{2}}{\sum_{l}\left(|h_{k}^{l}|\beta_{l}\sqrt{p_{l}}\right)^{2}} \right)\right]\\ &=\frac{1}{2}\log\left[1+\frac{P_{s}}{N}\left(|h_{k}^{s}|^{2}+\sum_ {l}|h_{l}^{s}|^{2}\right)\right]\end{split}  \tag{4}\]Mathematically, we can write the normalized sum-rate as:  
\[F_{1}=\frac{\sum_{k=1}^{K}C_{k}}{\sum_{k=1}^{K}C_{k}^{max}} \tag{5}\]  
The second objective is to reduce the carbon footprint or CO\({}_{2}\) emissions. The CO\({}_{2}\) emissions are measured in grams. If \(P\) is the power used in the transmission and \(X\) is a constant in grams/watt, then the product of \(P\) and \(X\) (_i.e._, \(PX\)) represents the CO\({}_{2}\) emissions in grams. The value of \(X\) is different for different types of material (fuel) used for electricity generation. There are three major sources of fuel for electricity generation. These fuels are oil, gas and coal. The value of \(X\) for lignite/brown coal, natural gas, crude oil and diesel oil is 940, 370, 640 and 670 grams/watt, respectively [11]. We define by \(E_{l}^{CO_{2}}=Xp_{l}\), the CO\({}_{2}\) emissions due to the \(l\)th relay, and \(E_{s,k}^{CO_{2}}=Xp_{k}^{s}\), the CO\({}_{2}\) emissions due to the source transmission power in the \(k\)th user band. We will normalize the CO\({}_{2}\) emissions objective function with \(E_{l^{max}}^{CO_{2}}+E_{s^{max}}^{CO_{2}}\), where \(E_{l^{max}}^{CO_{2}}=\sum_{l}Xp_{l}^{max}\) and \(E_{s^{max}}^{CO_{2}}=XP_{s}\). We can write the objective of CO\({}_{2}\) emissions as:  
\[F_{2}=\frac{E^{CO_{2}}}{E_{max}^{CO_{2}}} \tag{6}\]  
where \(E^{CO_{2}}=\sum_{l}E_{l}^{CO_{2}}+\sum_{k}E_{s,k}^{CO_{2}}\) and \(E_{max}^{CO_{2}}=E_{l^{max}}^{CO_{2}}+E_{s^{max}}^{CO_{2}}\). The joint objective of the GC-CRSN problem is to maximize the data-rate--_i.e._, \(F_{1}\)--and minimize the CO\({}_{2}\) emissions--_i.e._, \(F_{2}\). For joint optimization, we need to transform the objectives, \(F_{1}\) and \(F_{2}\), into a joint minimization (or maximization) objective.