# Elite Multi-Criteria Decision Making--Pareto Front Optimization in Multi-Objective Optimization
## 3 Methodology
### Algorithm  
The eMPF optimization technique combines EA with MCDM, as shown in Figure 2. Initially, a random population of \(T=(p+q)\) is generated and evaluated in the environment. The environment can be user defined as per requirements. With the fitness values evaluated from the simulator, the Pareto fronts of the population are generated as discussed in Section 2.2. This results in \(p\) better-performing individuals or policies over \(q\) individuals or policies.  
Once the Pareto fronts are generated, the number of individuals to remove from the required Pareto front set is determined on \(q\) individuals. If the number of individuals is less than or equal to 3, then any of those individuals can be selected. If the number of individuals is more than 3, then the TOPSIS method is applied on all the individuals of  
Figure 2: eMPF process flow chart.  
the Pareto front except for the extreme solutions, as shown in Algorithm 1. Before using the TOPSIS method, weights for the objectives need to be found. Entropy and degree of diversification are used to find weights for the objectives in this paper. The best outputs of TOPSIS will be given priority over other individuals for survival. Next, priority is given to extreme individuals. Finally, the priority of survival of individuals is given depending on the ranking derived from the TOPSIS method.  
```
Input: Pareto front \(P\), Number of individuals to select \(N\) Output: Selected individuals \(S\) if\(N\leq 3\)then \(S\leftarrow\) Randomly select \(N\) individuals from \(P\) else \(E\leftarrow\) Select individuals with the extreme fitness values from \(P\)  Calculate the weights of each objectives  Apply TOPSIS on individuals except \(E\) to rank individuals \(S\leftarrow\) Select the best individual, individuals with the lowest fitness, followed by individuals as per rank in TOPSIS endif return\(S\)
```  
**Algorithm 1** eMPF selection process  
Once the \(q\) policies are removed from the population using mutation \(q\), new individual policies are developed. The process of evaluation is repeated on \(p+q\) policies. The methodology for the research is shown in Algorithm 1.  
There are two major differences between the eMPF and M-PF optimizers. The first difference is the selection of extreme fitness values in the Pareto front. This gives an advantage to the eMPF optimizer of exploring the entire decision space over the M-PF optimizer, which helps the optimizer explore the search area in between the extreme fitness values. An extreme fitness value is the fitness value of an individual in the entire population with the lowest (in minimization) or highest (in maximization) objective value toward one objective function. The second difference is that in eMPF, the selection of individuals is based on the individuals within the Pareto front. In M-PF, the first Pareto front individuals are used for selection of the best individuals in the required Pareto front, whereas in eMPF, the individuals in the same Pareto front are used for selection of individuals. This transforms the optimizer from an indicator-based optimizer to a dominance-based optimizer. The improvements in the performance of the optimizer are shown in results section (Section 4).  
From the aforementioned second variation between the eMPF and M-PF optimizers, the time complexity is reduced for the eMPF optimizer. In M-PF, two loops, one each for the first Pareto front and the other required Pareto front, are needed. With eMPF, only one loop is necessary in the required Pareto front, reducing the overall time complexity with O(\(n\)), where \(n\) is the number of individuals in the first Pareto front.