## 1 Introduction
### Contributions and Organization  
In this paper, we investigate dual-hop CRSN that jointly maximizes the throughput and minimizes the total transmission power by assigning multiple relays to the users. In the future, we will investigate the optimal number of hops in CRSN to jointly maximize the throughput and minimize the total transmission power (or minimize the CO\({}_{2}\) emissions). We use a combination of shared-band non-regenerative amplify and forward relaying and cognitive radio to solve the problem of the crowded spectrum, the demand for a high data rate and global warming. Data rate maximization and power minimization are two conflicting objectives. Determining the optimal set of decision variables for a single objective, e.g., CO\({}_{2}\) emissions minimization, can result in a non-optimal set with respect to other objectives, e.g., sum-capacity (throughput) maximization. In our formulation, we use the normalized weighted sum method (WSM) to combine these conflicting objectives. In WSM, the weight of each objective is proportional to its importance, placed for decision-making. A WSM [35] without normalization would result in a biased fitness function--e.g., if the value of one objective function is in the range [0, 1] and the value of second objective is in the range [0, \(x\)] (where \(1<x\leq\infty\)), then the second objective produces bias in the weighted fitness function. In this work, we normalize all the objective values within the range [0, 1]. We formulate our problem in a way that the range of the combined objective function is always within 0 and 1. According to the best knowledge of the authors, there is no joint multiple relay assignment and power allocation scheme in the literature that deals with the analysis and optimization of the energy efficiency in a shared band multi-user cognitive radio system. The motivation of this work is to fill the gap, especially important for future green radio communications, with the aim of analyzing the shared-band multiple relay assignment and power allocation problem that maximizes the data rate and minimizes the CO\({}_{2}\) emissions. The main contributions of this paper are summarized as follows:  
1. We propose a multi-objective optimization framework that jointly exploits the crowded spectrum, the demand for a high data rate and global warming with the help of relay-assisted GC-CRSN. The proposed multi-objective framework jointly performs multiple relay assignment and power allocation in GC-CRSN, while optimizing two conflicting objectives. The first objective is to maximize the total throughput, and the second objective is to minimize the total transmission power of GC-CRSN. 2. For multiple relay assignment, we use a shared-band amplify and forward protocol. We also drive an upper bound on the data rate of the shared band amplify and forward protocol. This upper bound is useful for normalization in multi-objective optimization. 3. The proposed joint multiple relay assignment and power allocation problem is a non-convex mixed-integer non-linear optimization problem (NC-MINLP), which is generally NP-hard. We introduce a hybrid heuristic algorithm for this problem. The hybrid heuristic is a combination of the estimation-of-distribution algorithm (EDA) for performing power allocation and an iterative greedy algorithm for constraint satisfaction and relay assignment. 4. In addition to applying the EDA to the constrained multi-objective optimization problem for GC-CRSN, we also propose a modification in the EDA that greatly improves its performance. 5. A detailed analysis of the performance of the proposed algorithm is presented with the simulation results. We use \(\mathbf{A}\), \(\mathbf{a}\) and \(\mathbf{a}\) to represent matrix, vector and an element of a vector, respectively. When \(a_{i}\geq 0\) for all components, \(i\), of a vector, \(\mathbf{a}\), we use \(\mathbf{a}\geq 0\). Table 1 presents the summary of notations and symbols used in this paper. The rest of the paper is organized as follows. The system model is presented in Section 2.