# Shapley's Value as a Resource Optimization Strategy for Digital Radio Transmission over IBOC FM
## 3 Results
### Resource Optimization of the Proposed Scenario Using Linear Programming Techniques  
Linear programming has proven to be an invaluable tool in optimizing resources, such as bandwidth, by providing a systematic and efficient approach to the optimal allocation of these limited resources. Its mathematical methods enable precise modeling of the constraints and objectives of the problem, leading to optimal solutions that maximize system performance. In accordance with the proposed scenario, the linear programming problem can be formulated as follows:  
\[Maximize\sum_{i=1}^{n}x_{i} \tag{12}\]  
Subject to:  
\[0\leq x_{i}\leq d_{i}\]  
\[\sum_{i=1}^{n}x_{i}\leq BW_{T}\]  
where \(n\), \(d_{i}\), and \(x_{i}\) correspond to the number of nodes (n = 12), the bandwidth requested by node \(i\), and the bandwidth assigned to node i, respectively. The LP problem was solved  
Figure 4: BW requested and BW assigned (Shapley) for a channel saturation state.  
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using the Optimization Toolbox in MATLAB R2023a, employing various optimization methods. The objective function, constraints, and initial iteration point were organized in matrix form. The values were set as follows:  
_F:_ Vector of coefficients of the Objective function.  
_A,b:_ Inequality restrictions (\(Ax\leq b\)).  
\(BW_{T}:1150\) Kbps.  
_lb, ub:_ Set the lower and upper limits allowed for each of the nodes, respectively.  
Finally, the following expression is used to calculate the optimal solution to the problem using the \"interiorpoint-legacy\" algorithm, which yielded the best results compared to the \"dual-simplex\" and \"interior-point\" algorithms:  
options = optimoptions(\"linprog\",\"Algorithm\",\"interior-point-legacy\");  
[x,fval] = linprog(F,A,b,[],[],lb, ub, options)  
where _x and fval_ correspond to the solution vector and the maximum value that the objective function can reach. Figure 5 presents the results obtained for the proposed optimization model as a function of the bandwidth requested by each node.  
The data presented in Figure 5 represent the results of the optimization process using linear programming to allocate bandwidth to each node in the system. It is observed that the amount of bandwidth requested by each node is generally greater than the amount allocated, once again indicating the presence of resource constraints in the system. However, upon comparing the assigned bandwidth values, it can be seen that in this case, the nodes have received, on average, a slightly larger amount of bandwidth in some nodes compared to the results obtained in Figure 4. This difference may be attributed to variations in optimization approaches between linear programming and the Shapley value method.  
In terms of fairness in the distribution of resources, some variability is also observed among the nodes, suggesting that linear programming may prioritize the allocation of resources more uniformly among the nodes, aiming to maintain a balance in the system's performance and satisfy the needs of the different nodes in a more proportional manner.