Using Neural Networks To Predict Sets
The 75^6 option is not only bad for speed, but it is a very difficult representation to train, because the NN doesn't "understand" that any of the output categories are related. You would need an immense amount of data to train such a network, because ideally you need at least a few examples in any category that you expect the network to predict. Unless you had literally billions of examples to train from, the chances are certain combinations will never occur in your training set, thus could never be predicted with any confidence.

Therefore I would probably use 75 outputs, one for each object representing the probability that it would be chosen. This is easy to create training data for, if you have training examples with the 6 favoured objects - just a 1 for the objects chosen and 0 for all others as a 75-wide label.

For prediction, select the 6 objects with the highest probabilities. If these choices are part of a recommender system (i.e. may be presented to same person as being predicted for), then you can select items randomly using the outputs as weights. You may even find that this weighted Monte Carlo selection works well for predicting bulk user behaviour as well (e.g. for predictions fed into stock purchases). In addition, this stochastic approach can be made to predict duplicates (but not accurately, except perhaps averaged over many predictions).

A sigmoid transfer function on the output layer is good for representing non-exclusive probability. The logloss objective function can be used to generate the error values and train the network.

If you want to accurately predict duplicate choices out of the 6 items chosen, then you will need plenty of examples where duplicates happened and have some way to represent that in the output layer. For example, you could have double the number of output neurons, with two assigned to each object. The first probability would then be probability of selecting the item once, and the second probability would be for selecting it twice.



The question has since been updated, and it appears there are strong relationships between items making the choice of a set of items potentially very recipe-like. That may reduce the effectiveness of the ideas outlined above in this answer.

However, using 75 outputs may still work better than other approaches, and is maybe the simplest setup, so I suggest still giving it a try, even if just to establish a benchmark for other ideas. This will work best when decisions are driven heavily by the feature data available, and when in practice there are lots of valid choices for combining items so there is a strong element of player preference. It will work less well if there is a large element of game mastery and logic in player decisions in order to combine items.