The limitation is just the typical number of bits in an integer on 32-bit machines.

Let me reiterate Abigails sound advice, however. Say you have 100 checks. Then any exhaustive search will take 2**100 tries. That is around 10**30, so for 1 msec per try, the program would take 10**27 seconds. The age of the universe is only around 5 * 10**17 sec, so exhaustive search is hopeless.

Even creating a more clever branch-and-bound algorithm will only potentially reduce the factor of 2 a little.

So I think you need to rethink your task. Do you really need exact amounts, or can you approximate? If all the checks were even, but the desired amout was odd, there would be no solution; would your business collapse at that point? Look at what you can do to relax the constraints. There are fine polynomial heuristics for the knapsack problem that get you close in a modest time.