The simplest way to explain this is with an example.
You’re given then number 19 and you have a set of numbers to choose from: 1, 2, 3, 4, 6
When choosing values from the list (which can be duplicates), the smallest number of values that adds up to 19 is 4:
6, 6, 6, 1
6, 6, 4, 3
Now the list of numbers I want to select is 6, 6, 4, 3 because when plotted on a graph, it draws a smoother gradient going from larger numbers to smaller numbers. 6, 6, 6, 1 is a sudden drop as it goes from the final 6 to the 1.
I may be wrong (and please correct me if so) but I think a better way of expressing that desire is to choose the list where the product of each of the numbers is the largest value.
6 * 6 * 6 * 1 = 216
6 * 6 * 4 * 3 = 432 (choose this list)
Another example is choosing 4, 3 instead of 6, 1 for a value of 7.
How could this be calculated using an algorithm (ideally non-brute-force!) for use with any total value and an arbitrary set of numbers to choose from?
This is the subset sum problem. The subset sum problem is NP-complete. In general the solution requires an exhaustive search, so the run time complexity is O(2^N) where N is the number of values in the set. Alternatively, if the desired sum S is small, dynamic programming can be used with a run time complexity of O(NS).