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I have 9 bins named A through I containing the following number of objects:

A(8), B(7), C(6), D(7), E(5), F(6), G(6), H(6), I(6)

Objects from each bin fulfill a specific role and cannot be interchanged. I am selecting one object from each bin at random forming a "team" of 9 "players":

T_ijklmnopq = {a_i, b_j, c_k, d_l, e_m, f_n, g_o, h_p, i_q}

There are 15,240,960 such combinations - a huge number. I have means of evaluating performance of each "team" via a costly objective function, F(T_ijklmnopq). Thus, I can feasibly sample a limited number of random combinations, say no more than 500 samples.

Having results of such sampling, I want to predict the most likely best combination of "players". How to do it?

Keep in mind this is different from classical team selection because there is no meaningful evaluation of F() based on individual performance. For example, "player" a_6 may be good individually, but he may not "like" e_2 and therefore the performance of "team" containing the two suffers. Conversely, three mediocre players b_1, f_5, i_2 may be a part of an awesome "team". What's know is the whole "team" performance, that's all.

One more detail: contributions of the individual roles A through I are not weighted equally. Position of, say, E may be more important than, say, H. Unfortunately, these weights are not known upfront.

The described problem must be know to combinatorial analysts, but I haven't found anything exactly like it. Linear programming solutions with known individual "player" scores do not apply here. I will be most grateful for a specific name under which this problem is known to experts.

So far I have collected 400 samples. Here is a graph of the sorted F(T) values vs. a (arbitrary) sample number to illustrate that F(T) is "reasonable". F(T) graph of sorted samples

user3002498
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    In order to avoid having to brute-force everything, you'd need to have some sort of reasonable assumptions about how T can change as you swap players in or out. Otherwise, you could imagine a T function that is zero for all teams except for one team, where it's huge, and in that case you'd have no way of figuring out the best team without exhaustively searching all options. So are there any reasonable limits you can place on how much the value can change as you add or remove someone? – templatetypedef Oct 25 '22 at 22:13
  • I have collected 396 samples so far and the F(T) changes are quite reasonable and smooth. The zero-to-huge scenario you described is highly unlikely and the F(T) does follow reasonable limits, indeed. – user3002498 Oct 25 '22 at 23:53
  • @user3002498 without more information about F(T), I don't think we'll be able to give too much advice. I think the best you'll be able to do is randomly sample some ~100 points, and then try walking the "best scoring" towards local maxima. Look up gradient descent – Dillon Davis Oct 26 '22 at 00:01
  • Hi Dillon, I have just added a graph of 400 samples sorted by F(T). What do you mean by gradient descent? The inputs to F(T) are discrete labels... I am mainly interested if this kind of problem is known to mathematicians (I suspect it is). If yes, then under what name. – user3002498 Oct 26 '22 at 00:40

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