In the late 60-ties my boss formulated a technical problem: “Try to find system solutions that are insensitive to variations in parameter or component values due to the statistical spread in manufacturing” he said. This means that he wanted the manufacturing yield maximized.

In the middle of the 60-ties, I worked at a Swedish telephone company with analysis and optimizations of signal processing systems. Formerly such systems consisted of interconnected components such as resistors, inductors and capacitors.

In the late 60-ties my boss formulated a technical problem: “Try to find system solutions that are insensitive to variations in parameter or component values due to the statistical spread in manufacturing” he said. This means that he wanted the manufacturing yield maximized.

If we have only two components – each having a parameter value – the problem is very simple. Let the first parameter value be the shortest distance to the left edge of a picture (below) while the second value is the distance to the bottom edge. Then, if the interconnection is given, a point in the picture represents the system unambiguously.

Suppose now that all points inside a certain triangle (region of acceptability, marked by red edge) will meet all requirements according to the specification of the system, while all other points does not, and that the spread of parameter values is uniformly distributed over a circle (green). Then, if the circle touches the three sides of the triangle, the centre of the circle would be a perfect solution to the problem.

http://picasaweb.google.com/gregor744/GA_figures02?authkey=Gv1sRgCNLYgpOK2ZH_sQE#5392019720626734018

But if we have 10 or 100 parameters, then the number of possible parameter combinations becomes super-astronomical and the region of acceptability will not possibly be surveyed. I begun to think that the man was not all there.

The problem was almost forgotten until a system designer entered my room about half a year later. He wanted to maximize the manufacturing yield of his system that was able to meet all requirements according to the specification, but with a very poor yield.

Oh, dear! I would not like to get fired immediately. So, we wrote a computer program in a hurry, using a random number generator giving normally Gaussian distributed numbers. The system functions of each randomly chosen system were calculated and compared with the requirements. In this way we got a population (generation) of about 1000 systems from which a certain fraction of approved systems was selected. For the next generation the centre of gravity of the normal distribution was moved to the centre of gravity of the approved systems and this process was repeated for many generations.

After about 100 generations the centres of gravity reached a state of equilibrium. Then the designer said “but this looks very god”. And we were both astonished, because we had only put some things together by chance. A closer look revealed that there is a mathematical theorem (the theorem of normal or Gaussian adaptation), valid for normal distributions only, stating:

“If the centre of gravity of the approved systems coincides with the centre of gravity of the normal distribution in a state of selective equilibrium, then the yield is maximal.”

This gave an almost religious experience. Here a mathematical theorem solved our problem without our knowledge and independently of the structure of the region of acceptability.

Our very simple process was similar to the evolution of natural systems in the sense that it worked with random variation and selection. Later, it turned out that evolution might as well make use of the theorem.

Gkm