For efficient algorithm on "minimization" knapsack problem
There's a simple reduction from knapsack. Binary search for the solution to your knapsack instance, then solve the "dual knapsack" with that value as your covering constraint $B$. Compare the value given by the "dual knapsack" against your knapsack packing constraint, which gives you the direction to continue the binary search.

I think you can use the same rounding approach as the one for knapsack in order to get a PTAS for this problem. These notes notes should help you do it.