Optimization of multiple criteria: Pareto efficiency and fast heuristics should be more popular than they are

E veryone has learned in secondary school mathematics how to find the maximum or the minimum of a function and therefore, optimization is considered a simple task by the majority of people including scientists. Lagrange multipliers are well known as the tools to introduce constraints. An insider, however, knows that textbook knowledge helps only very little in real world problems and finding solutions for more involved optimization tasks is commonly quite tough. A recently published collective volume [1] reports the results of a study group of the Berlin-Brandenburg Academy of Sciences on Structure Evolution and Innovation, which among other things is dealing with the problem of optimization in case of multiple goals and how humans can successfully manage such complex situations. It is self-evident that multiple criteria lead to conflicting situations unless they are independent, and complete independence is quite uncommon in reality. We illustrate by means of a simple example: Driving a car from point A to point B takes time and consumes gasoline. Clearly, everybody wants to reach B as quickly as possible and the gasoline costs should be minimal. Every car requires more gasoline to drive faster and so we are dealing with two naive pseudo-optima: (i) the shortest time to reach B from A and (ii) the most economic speed to drive from A to B, and needless to say these two optima will be different. Comparison of cars from different car producers is of interest for every car buyer. If the customer putting other criteria aside is only interested in potential speed and gasoline costs a survey of price and performance can be made by means of a simple diagram showing Pareto efficiency. These two criteria–speed versus economy–are plotted in Figure 1, where we idealized somewhat as we assumed continuous variation and smooth curves: The products of most car fabricants in this particular plot are close to the Pareto optimum that appears as an insurmountable hyperbola like frontier between the possible and the impossible. One product shown as a gray star in the figure is Pareto inefficient because its performance can be improved with respect to both criteria keeping the efficiency constant with respect to the other criterion, that is, reducing fuel consumption without changing maximal speed or increasing maximal speed without changing costs. A single optimal point can be achieved through the consideration of individual preference, which defines a tangent expressing, for example, how much money PETER SCHUSTER