## 4 Design and Implementation of the Optimization Framework
### Multiobjective GA for Pareto-Front Optimization  
Despite the fact that the evaluation by a weighted sum cost function facilitates the optimization process to a great extent, the most significant limitation of this process is that the metrics (e.g., energy consumption-_mJ_, latency-_ms_, packet loss-_how many_) used in the cost function formula are sometimes not in the same scale. Even with the use of probability values in the experimental part, the equality of scale and avoidance of implicit weighting cannot be strictly guaranteed, considering that if the given energy volume for each node is set to be a large value (e.g., 10,000 _mJ_) or that if a high maximum latency is used as the dividend, then the importance of energy and latency impacts will be reduced. A more suitable solution to consider trade-offs among different metrics is Pareto-front based multi-objective optimization. As mentioned previously in Section 4.3, simultaneous multi-objective (Equation (1)) based techniques rarely exist in conventional optimization methods. However, the GA based multi-objective optimization provides an alternative, since the parallelizable characteristics of GA (represented by the population size) can help evaluate many different sets of solutions in the parameter space simultaneously, which greatly improves the efficiency. The selection of the best individuals is based on the Pareto-front (\(\partial F\)\(\prec\), \(F\)-solution space), which can be described as a solution \(\mathbf{f}=\)[\(f_{1}\),...\(f_{n}\)] dominating \(\mathbf{f}^{*}=\)[\(f_{1}^{*}\),...\(f_{n}^{*}\)]. This condition is true if each parameter of \(\mathbf{f}\) is not greater than the corresponding parameter in \(\mathbf{f}^{*}\) and there is at least one parameter that is less, _i.e._, \(\mathbf{f}_{i}\leq\mathbf{f}_{i}^{*}\) for each \(i\) and \(\mathbf{f}_{i}<\mathbf{f}_{i}^{*}\) for some \(i\). This is presented as \(\mathbf{f}\sim\mathbf{f}^{*}\) to mean \(\mathbf{f}\) dominates \(\mathbf{f}^{*}\), and the total description in mathematics can be given as follows:  
\[\forall_{(i\in\{1,\ldots n\})}\ (f_{i}\leq f_{i}^{*})\ \ \wedge\ \ \exists_{(i\in\{1,\ldots n\})}\ (f_{i}<f_{i}^{*}) \tag{5}\]  
In other words, the Pareto-front is part of the boundary of performance space. On this boundary, no solution is better in criteria that makes at least one performance metric better without making other metrics worse. The Pareto front captures the trade-offs between competing performance metrics and identifies the solutions that are non-dominated, as shown in above Figure 3.  
In addition, with design problems becoming more complex and the system under investigation integrating multiple functionalities, performance metrics of several different cases need to be evaluated at the same time. This cannot be achieved if multi-objective optimization only focuses on a specific condition. Therefore, a multi-scenario optimization method is proposed and already has applications in  
Figure 3: Pareto-front optimality. many engineering problems such as [22, 38]. The use of this multi-scenario based multi-objective optimization is to find a robust solution which can give the optimal performance for all possible case scenarios and it can be presented as:  
\[\min_{(X)}\textbf{p}=[\underbrace{p_{\alpha 1},p_{\alpha 2},...,p_{\alpha }}_{\textbf{Scenario\_a}},......,\underbrace{p_{m1},p_{m2},...,p_{m}}_{\textbf{ Scenario\_m}}] \tag{6}\]  
where \(X\) is the vector of design parameters, and **p** denotes a set of related performance metrics of several scenarios which are returned by iMASKO evaluation engine from the simulations. Take two primary objectives, energy and reliability (packet loss), as the example in the optimization of network performance. If these two metrics need to be evaluated under different scenarios before practical use, the following four-dimensional (4D) front in above Figure 4 is able to transpose the concept for the case of multiple scenarios and help explore trade-offs for the metrics.