Sorting

PEPSO uses the non-dominated sorting method. In this method, at first, objective values of all solutions will be compared to find the number of times that other solutions dominated each solution. By definition, solution A dominates solution B if both of these conditions are true: 1) the solution A is no worse than B in all objectives and 2) the solution A is strictly better than B in at least one objective (Deb 2001). Mathematical definition of domination for a minimization problem has been presented by equation 6 (Narzisi 2008):

𝐴 ≼ 𝐵 𝑖𝑓𝑓 {𝑓𝑖(𝐴) ≤ 𝑓𝑖(𝐵) ∀𝑖 ∈ 1, … , 𝑀

∃𝑗 ∈ 1, … , 𝑀 𝑓_𝑖(𝐴) < 𝑓_𝑖(𝐵) Equation 6

Where: A and B are two solutions,

𝑓_𝑖(𝐴) is value of 𝑖𝑡ℎ _{objective of solution A and}

𝑀 is number of objectives of the minimization problem

Those solutions that have not been dominated by other solutions will be placed in the first Pareto frontier (Rank 1). Similarly, those solutions that are dominated just once will be placed in the second Pareto frontier (Rank 2), and so on. Figure 19 helps to visualize the idea of non-domination ranking for a two objective solution space. In this figure, “X1” and “X2” axis show values of two objectives of each solution that have been shown by different markers. For instance, “X1” and “X2” objective values of the solution “A” are 45 and 30, respectively. Comparing objective values of solution “A” and “B” suggest that “X1” objective of solution “B” is smaller than solution “A”. Also, the “X2” value of objective “A” is smaller than “B”. This problem is a minimization optimization problem with the utopia point of (0,0), So solution “A” is better than solution “B” in respect to “X2” objective but is worse than solution “B” in respect to “X1” objective. So none of these two solutions has an absolute advantage over the other, and neither dominates the other. Both solutions have been placed on the same Pareto frontier - as shown by the rectangular orange markers in Figure 19. However comparing solution “A” and “C” shows that Solution “A” is better than solution “C” on both objectives. So solution “C” is dominated by solution “A” and cannot be put on the same Pareto frontier as solutions “A” and “B”.

After non-domination ranking and finding the rank of each solution based on its Pareto frontier rank, the crowding distance of solutions that are within the same Pareto frontier will be calculated. The value of the crowding distance is used to sort solutions within a Pareto frontier (those that have the same rank). By convention, crowding distance of

solutions that are located on edges of Pareto frontier is infinity. The first step in the calculation of crowding distance is the sorting (in ascending manner) of the solutions based on values of one objective. Then the solution with minimum objective value will be selected as the edge of the Pareto frontier, and its crowding distance will be infinity. The crowding distance of next solutions can be calculated by Equation 7 (Deb 2001).

𝐶𝐷_𝑖(𝑆_𝑗) =𝑓𝑖(𝑆𝑗+1)−𝑓𝑖(𝑆𝑗−1)

𝑓_{𝑖𝑚𝑖𝑛}−𝑓_{𝑖𝑚𝑎𝑥} Equation 7

Where: 𝐶𝐷_𝑖(𝑆_𝑗) is crowding distance of 𝑗𝑡ℎ solution in the sorted Pareto frontier based on 𝑖𝑡ℎ

objective

𝑓_𝑖(𝑆_𝑗+1) and 𝑓_𝑖(𝑆_𝑗−1) are 𝑖𝑡ℎ _{objective values of a solution before and a solution after the 𝑗}𝑡ℎ _solution

in the sorted Pareto frontier based on 𝑖𝑡ℎ objective

𝑓_{𝑖 𝑚𝑖𝑛} and 𝑓_{𝑖 𝑚𝑎𝑥} are smallest and largest values of the 𝑖𝑡ℎ objective among solutions of the Pareto Frontier

In the end, the same process will be repeated based on values of other objectives. The calculated crowding distance values of all objectives of a solution will be summed to provide the total crowding distance of the solution.

Figure 19 provides further insight into the crowding distance calculation. Figure 19 shows solutions that have been categorized in four Pareto frontiers. Solutions of each Pareto frontier have been shown with the same color and same marker shape. Solutions of the first Pareto frontier are shown by blue circle markers. Solutions D and E are two edges of the first Pareto frontier with minimum X1 and X2 values respectively. By definition, crowding distance value of these two solutions is infinity. Crowding distance of other solutions of the first Pareto frontier can be calculated by Equation 7:

Crowding Distance of Solution F = [(75-30) / (75-0)] + [(25-0) / (70-0)] = 0.957 Crowding Distance of Solution G = [(20-10) / (75-0)] + [(50-40) / (70-0)] = 0.276

As the crowding distance of solution F is larger than crowding distance of solution G, solution F will have a higher rank in respect to solution G among the solutions of the first Pareto frontier. After calculating the crowding distance values, solutions can be sorted initially based on the rank of their Pareto frontiers and subsequently based on their crowding distance within each Pareto frontier. By using this sorting method, all solutions of Figure 19 have been sorted and their rank are shown with two numbers that are separated by a dash (“-“). The first number shows Pareto frontier rank of each solution and the second number shows the rank of the solution among the same Pareto frontier. So Solution D and E from the edges of the first Pareto frontier have the highest rank in the whole population and solution H with the lowest crowding distance in the last Pareto frontier has the lowest rank. This sorting method helps the optimization algorithm to put

more value on solutions that have been dominated less and are located in less crowded (less explored) regions of the solution space or on the edges of Pareto frontiers.