In the $0-1$ Knapsack problem, we are given a set $S$ of items and the goal is to find a subset $T$ of $S$ that does not exceed the capacity $W$ and maximizes the total value of the items in $T.$ Now consider the following generalization of the problem. As before, we have a knapsack with integer capacity $W,$ and we have a set $S$ of items, each with weight $w_i$ and value $v_i,$ but we have a limited number of copies $($call it c$)$ of such items available to us$.$We want to know how to optimally fill the knapsack.

a$.$ Formulate this variation of the $0-1$ Knapsack problem as a computational problem. That is$,$ specify the input and output of the problem. $($This is not as easy as it first seems. In particular, what should the output be$?)$

b$.$ Design a dynamic programming solution $($write pseudocode$)$ to this problem. Your pseudocode can be a combination of high$-$level and low$-$level pseudocode. $($Hint: Think about the dynamic programming solution to the $0-1$ Knapsack problem. What possibilities do we now need to consider in this generalized version of the problem?$)$

c$.$ What is the worst case running time of your algorithm?