# 8 f f g i 9 break out of inner for loop 10 else if g

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8:FF-{Gi}9:breakout of inner for-loop10:else ifGiPareto DominatesFjgivenOthenAn existing front member knocked out!11:FF-{Fj}12:returnF132
Computing the ranks is easy: figure out the first front, then remove the individuals, then figureout the front again, and so on. If we pre-process all the individuals with this procedure, we couldthen simply use the Pareto Front Rank of an individual as its fitness. Since lower Ranks are better,we could convert it into a fitness like this:Fitness(i) =11+ParetoFrontRank(i)The algorithm to compute the ranks builds two results at once: first itpartitionsthe populationPinto ranks, with each rank (a group of individuals) stored in the vectorF. Second, itassignsarank number to an individual (perhaps the individual gets it written internally somewhere). Thatway later on we can ask both: (1) which individuals are in ranki, and (2) what rank is individualjin? This procedure is calledNon-Dominated Sorting, by N. Srinvas and Kalyanmoy Deb.106Algorithm 101Front Rank Assignment by Non-Dominated Sorting1:PPopulation2:O{O1, ...,On}objectives to assess with3:PPWe’ll gradually remove individuals fromP4:RInitially empty ordered vector of Pareto Front Ranks5:i16:repeat7:RiPareto Non-Dominated Front ofPusingO8:foreach individualARido9:ParetoFrontRank(A)i10:PP-{A}Remove the current front fromP11:ii+112:untilPis empty13:returnRCheaperMore Energy EfficientABB1B2A1A2Figure 51The sparsity of individualBishigher than individualAbecauseA1+A2<B1+B2.SparsityWe’d also like to push the individuals in the pop-ulation towards being spread more evenly across the front.To do this we could assign a distance measure of some sortamong individuals in the same Pareto Front Rank. Let’sdefine thesparsityof an individual: an individual is in amoresparse regionif the closest individuals on either sideof it in its Pareto Front Rank aren’t too close to it.Figure 51 illustrates the notion we’re more or less after.We’ll define sparsity asManhattan distance,107over everyobjective, between an individual’s left and right neighbors106First published in N. Srinivas and Kalyanmoy Deb, 1994, Multiobjective optimization using nondominated sortingin genetic algorithms,Evolutionary Computation, 2, 221–248. This paper also introduced Algorithm 100.107Manhattan lies on a grid, so you can’t go directly from point A to point B unless you’re capable of leaping tallbuildings in a single bound. Instead you must walk horizontally so many blocks, then vertically so many blocks. That’sthe Manhattan distance from A to B.133
along its Pareto Front Rank. Individuals at the far ends of the Pareto Front Rank will be assignedan infinite sparsity. To compute sparsity, you’ll likely need to know the range of possible valuesthat any given objective can take on (from min to max). If you don’t know this, you may be forcedto assume that the range equals 1 for all objectives.

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