Power Magnetic Devices. Scott D. Sudhoff

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with ε = ε1a yields x1a with objective function values (ε1a,f2(x1a)) which is a point on the Pareto‐optimal front, as shown in Figure 1.17. Solving (1.7‐2) with ε = ε1b yields x1b with objective function values (ε1b,f2(x1b)). Repeating the procedure over a range of ε values, the Pareto‐optimal set and front can be determined.

Schematic illustration of Calculation of Pareto-optimal front with ε-constraint method.

      In more concrete terms, suppose we wished to minimize volume and loss for some electromagnetic device. If our first objective was volume and our second objective was loss, we would repeatedly minimize loss with different constraints on the volume. In this case, each ε value would correspond to a different volume. For each numerically different volume constraint, we would get the corresponding loss and thereby—one single‐objective optimization at a time—build up a Pareto‐optimal front between volume and loss.

      As it turns out, GAs are well‐suited to compute Pareto‐optimal sets. This is because they operate on a population. In multi‐objective optimizations, the population of designs can be made to conform to the Pareto‐optimal set. This allows a GA to determine the Pareto‐optimal set and Pareto‐optimal front in a single analysis without requiring the solution of a separate optimization for every point on the front as is needed in the weighted sum method, ε‐constraint method, and weighted metric methods.

      GAs are well‐suited for multi‐objective optimization, and there are a large number of approaches that can be taken. The goal in all of these methods is to evolve the population so that it becomes a Pareto‐optimal set.

      Multi‐objective GAs fall into two classes, nonelitist and elitist. The elitist strategies are particularly effective because they explicitly identify and preserve, when possible, the nondominated individuals. Elitist strategies include the elitist nondominated sorting GA (NSGA‐II), distance‐based Pareto GA, and the strength Pareto GA [11]. In order to keep the present discussion limited that we might soon start discussing device design, we will somewhat arbitrarily focus our attention on the elitist NSGA‐II.

function R=front(S) if ∣S∣=1 R=S else i=floor(∣S∣/2) T=front(S1:i) B=front(Si+1|S|) N=solutions of B not dominated by any solution of T R=T ∪ N end end

      (1.8-1)equation

      In executing this example, calls 4, 5, 7, 8, 11, 12, 14, and 15 return nondominated sets because their input and output argument set size is 1. In calls 3, 6, 10, and 13, no member of an input set T can be dominated by the corresponding set B because of the ordering with respect to the first objective. The return arguments of calls 3, 6, 10, and 13 cannot have dominated members because the input sets to these calls (T and B) were nondominated, no member of T can be dominated by a member of B, and every member of B is checked against every member of T. Thus, proceeding from the bottom of Figure 1.18 to the top as subroutine calls are made, we see that the process involves a gradual combining and sifting of smaller nondominated sets to yield a larger nondominated set.

Schematic illustration of application of Kung’s method.

      In addition to Kung’s method, two more processes will be needed in order to set forth the elitist NSGA‐II. The first of these is nondominated sorting (NDS). In order to understand NDS, let us consider again Figure

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