Search Space for Population Metaheuristic Optimization Problem

Given that x objects (repetition allowed) can be arranged into y configurations, and that the population size is N, I am uncertain as to whether the size of the search space is |S| = |(y^x) * N| or |S| = |(y^x)^N|.

|y^x| comes from basic combinatorics: it is the number of ways to arrange |x| objects into |y| configurations.

As an example, suppose x = (1, 2, 3, 4) and y = (1, 2, ... 5) and N = 100. Clearly, |x| = 4 and |y| = 5. The xs are drawn from some probability distribution (e.g., uniform). A configuration might look like

1 1 3 2 4

Another might be

4 3 1 1 3

for some previously-specified probability distribution. The underlying metaheuristic algorithm generates N = 100 such configurations randomly.

I am just unsure on how to incorporate N.

• I'm not sure what you mean by "arrange x objects into y configurations". I'm not sure what you mean by "population size". I suggest you edit the question to describe what you mean in more detail. You might need multiple sentences to specify the problem. It might be useful to break it down and define your terms (what's a configuration?).
– D.W.
Commented May 4, 2020 at 5:57
• Thanks. I will clarify. Commented May 4, 2020 at 6:02
• I have now added an illustrative example to my post to better explain the problem. Please let me know if it's clearer now Commented May 4, 2020 at 6:16

There are $$|x|^{|y|}$$ configurations. Then you want to generate $$N$$ configurations. By the same reasoning, there are $$(|x|^{|y|})^N$$ ways to do that.