Better algorithm for a multi-variable gambling puzzle than brute force? -

i don't know name describe problem, i'm going set example of , ask algorithm better brute force solve.

there 6 buckets, each 1 different size, each 1 holding different starting volume. each bucket has link prize value - call money - dispensed when bucket full.

there 8 valves. each valve feeds different subset of buckets when opened, , each bucket can different amount added when valve opened. each valve has fixed cost turn.

partial example: bucket 1: 14 of 20 gallons. pays $2.00 bucket 2: 12 of 25 gallons. pays $1.50 bucket 3: 18 of 35 gallons. pays $4.00 etc...

valve 1: costs $0.50. puts 2 gallons in 2 , 1 gallon in 3. valve 2: costs $0.40. puts 3 gals in 1 , 5 gals in 2 etc...

so pay money turn valve, , if fill bucket win money.

objective: calculate best order of valves turn net profit when buckets fill up, if profitable combinations exist.

when bucket full, resets pre-set starting volume (which may different current volume - buckets retain state valve turn valve turn). theoretically there no limit number of resets 1 potentially turn valves forever.

the obvious solution here brute force. reset variables start position, turn 1 valve, test output, reset, iterate next valve, etc. after 8 turned, start turning 2 valves (11, 12, 13, 14, ... 25, 26, 27, 28). every time find combination fills @ least 1 bucket, calculate net gain/loss , compare against previous results determine best.

slow , crude. direct, slow , crude.

so:

1. what name of type of problem? and
2. is there known algorithm helps solve more efficiently?

to clear - math specifics aren't bit need improving - can calculate result of sequence of valve operations regardless of minor variations in math.

it's brute force iteration want avoid. not want 8 single operations (v1, v2, v3...), 64 double operations (v1v1, v1v2, v1v3, etc...), 512 triple operations (v1v1v1, v1v1v2, v1v1v3) if can avoid it.