Weighted sum and penalty functions

When optimizing with the weighted sum, the single values of the ${\displaystyle O}$ objectives are first normalized so that they can be compared. This can be done with the help of costs or by specifying target values and determining the current value as the degree of fulfillment. Costs or degrees of fulfillment can then be compared with each other and, if required, can also be mapped to a uniform fitness scale. Without loss of generality, fitness is assumed to represent a value to be maximized. Each objective ${\displaystyle o_{i}}$ is assigned a weight ${\displaystyle w_{i}}$ in the form of a percentage value so that the overall raw fitness ${\displaystyle f_{raw}}$ can be calculated as a weighted sum:

${\displaystyle f_{raw}=\sum _{i=1}^{O}{o_{i}\cdot w_{i}}\quad {\mathsf {with}}\quad \sum _{i=1}^{O}{w_{i}}=1}$

A violation of ${\displaystyle R}$ restrictions ${\displaystyle r_{j}}$ can be included in the fitness determined in this way in the form of penalty functions. For this purpose, a function ${\displaystyle pf_{j}(r_{j})}$ can be defined for each restriction which returns a value between ${\displaystyle 0}$ and ${\displaystyle 1}$ depending on the degree of violation, with the result being ${\displaystyle 1}$ if there is no violation. The previously determined raw fitness is multiplied by the penalty function(s) and the result is then the final fitness ${\displaystyle f_{final}}$:^[13]

${\displaystyle f_{final}=f_{raw}\cdot \prod _{j=1}^{R}{pf_{j}(r_{j})}=\sum _{i=1}^{O}{(o_{i}\cdot w_{i})}\cdot \prod _{j=1}^{R}{pf_{j}(r_{j})}}$

This approach is simple and has the advantage of being able to combine any number of objectives and restrictions. The disadvantage is that different objectives can compensate each other and that the weights have to be defined before the optimization.^[12] In addition, certain solutions may not be obtained, see the section on the comparison of both types of optimization.

Pareto optimization

A solution is called Pareto-optimal if the improvement of one objective is only possible with a deterioration of at least one other objective. The set of all Pareto-optimal solutions, also called Pareto set, represents the set of all optimal compromises between the objectives. The figure below on the right shows an example of the Pareto set of two objectives ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ to be maximized. The elements of the set form the Pareto front (green line). From this set, a human decision maker must subsequently select the desired compromise solution.^[12] Constraints are included in Pareto optimization in that solutions without constraint violations are per se better than those with violations. If two solutions to be compared each have constraint violations, the respective extent of the violations decides.^[14]

It was recognized early on that EAs with their simultaneously considered solution set are well suited to finding solutions in one run that cover the Pareto front sufficiently well.^[14][15] Besides the SPEA2,^[16] the NSGA-II^[17] and NSGA-III^[18][19] have established themselves as standard methods.

The advantage of Pareto optimization is that, in contrast to the weighted sum, it provides all alternatives that are equivalent in terms of the objectives as an overall solution. The disadvantage is that a visualization of the alternatives becomes problematic or even impossible from four objectives on. Furthermore, the effort increases exponentially with the number of objectives.^[13] If there are more than three or four objectives, some have to be combined using the weighted sum or other aggregation methods.^[12]

Comparison of both types of assessment

Relationship between the Pareto front and the weighted sum. The set of feasible solutions ${\displaystyle Z}$ is partially bounded by the Pareto front (green).^[13]

Example of a non-convex Pareto front^[13]

With the help of the weighted sum, the total Pareto front can be obtained by a suitable choice of weights, provided that it is convex.^[20] This is illustrated by the adjacent picture on the left. The point ${\displaystyle {\mathsf {P}}}$ on the green Pareto front is reached by the weights ${\displaystyle w_{1}}$ and ${\displaystyle w_{2}}$, provided that the EA converges to the optimum. The direction with the largest fitness gain in the solution set ${\displaystyle Z}$ is shown by the drawn arrows.

In case of a non-convex front, however, non-convex front sections are not reachable by the weighted sum. In the adjacent image on the right, this is the section between points ${\displaystyle {\mathsf {A}}}$ and ${\displaystyle {\mathsf {B}}}$. This can be remedied to a limited extent by using an extension of the weighted sum, the cascaded weighted sum.^[13]

Comparing both assessment approaches, the use of Pareto optimization is certainly advantageous when little is known about the possible solutions of a task and when the number of optimization objectives can be narrowed down to three, at most four. However, in the case of repeated optimization of variations of one and the same task, the desired lines of compromise are usually known and the effort to determine the entire Pareto front is no longer justified. This is also true when no human decision is desired or possible after optimization, such as in automated decision processes.^[13]