Crowding distance

After the appropriate JG operation is completed (in box D, Fig. 4.5), the Np better parents (present in box P") and all the Np daughters (after crossover, mutation and the JG operations present in box D ) are copied into a new box, PD, having 2Np locations. These 2Np chromosomes are re-classified into fronts (in box PD ), using the concept of domination. The best Np chromosomes are taken from box PD and put into box P" (of size Np), front-by-front. In case only a few members are needed from the final front to fill up this box, the concept of crowding distance is used. It is clear that this procedure, called elitism, collects the best members from the parents as well as the daughters. 

Find the crowding distances (Idist,b of all As points in File 2 (as in NSGA-II)... 

Select NP individuals for the next generation, use crowding distance, if required. 

From Equation (6), it can be inferred that an individual with the lower rank is selected while if two points are in the same fi ont, the individual with the larger local crowding distance is selected. 

Replacement Combine the parent set and the offspring set, and then rank the combined set based on the fast-non-dominated sort selection and the crowding distance calculation. The new parent population is created by replacing it with the best set of solutions in the combined set. 

Filling all P by F to Fj, (from the best solutions in front 1) if there are more solutions in the same front (e.g., F ), then they are selected according to the rank of crowding distance as described in Deb (2009). Step 2 continues until a new set P" has been created. 

Multi-Objective Differential Evolution (MODE). This is a custom multiobjective variant (with elitism) of Differential Evolution (DE) [24], that simply combines with the classic DE mutation/crossover operators the non-dominated sorting and crowding distance mechanisms used in NSGA2 (see below). We set crossover rate Cr = 0.3 and scale factor F = 0.5. 

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