When your values - your a_n - can be negative, and your b (the goal) is zero, then it's called "subset sum". If the a_n are non-negative (i.e., some of the a's may be zero), the b is positive, and the choice is to either exclude (0) or include (1) one "copy" of each value, then it's a "0-1 knapsack" problem. Usually, but not always, a knapsack problem has components with multiple values, and the goal is a minimax problem: maximize the a's, while minimizing the c's.
It's always darkest, just after the lights go out.
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Subset sum or 0-1?
When your values - your a_n - can be negative, and your b (the goal) is zero, then it's called "subset sum". If the a_n are non-negative (i.e., some of the a's may be zero), the b is positive, and the choice is to either exclude (0) or include (1) one "copy" of each value, then it's a "0-1 knapsack" problem. Usually, but not always, a knapsack problem has components with multiple values, and the goal is a minimax problem: maximize the a's, while minimizing the c's.
It's always darkest, just after the lights go out.