Best algorithm for maximisation with two criteria


I am looking for the optimum algorithm for the following:

  • 10-15 players
  • Each player has between 20 and 40 cards.
  • Each card can have one of up to 200 possible characters, and a separate numerical rating (higher is better). Card’s characters may duplicate between players, or in players hands. Ratings are highly unlikely to exactly duplicate, though they may be close.

I need to select 5 ‘active’ cards from each players’ hands to meet the following criteria:

  1. All characters must be unique – no duplicates in any of the ‘active’ cards of any of the players, or across all players.
  2. The total of the players’ active cards’ ratings must be as high as possible.

Right now I:

1) go through all players and find the highest rated card still available; 2) mark it as active for the player whose card it is; and 3) mark that character as used for all other players (so it doesn't get used again). 

Repeat 1-3 until all players have 5 characters

This gives a pretty good result. But what if we had the following:

Player A: Character 1, rating 100 Character 2, rating 99 Character 3, rating 98  Player B: Character 1, rating 97 Character 4, rating 2 Character 5, rating 1 

For the sake of the example, assume we only need two active cards per players.

If Player A uses Character 1 per my algorithm then:

  • Player A: Character 1 + 2 = rating 199
  • Player B: Character 4 + 5 = rating 3
  • Total rating 201

Instead if Player A doesn’t use Character 1 then:

  • Player A: Character 2 + 3 = rating 197
  • Player B: Character 1 + 4 = rating 99
  • Total rating 296

So my algorithm does not produce the best team total rating.

Can anyone suggest a better algorithm, other than just brute force trying all the possible combinations to find the highest total rating? I wonder, for example, if there’s something about finding the optimum ratings for each player and then adjusting them to avoid duplication with other players; or perhaps something completely different.