Operations research - summation notation

Question

Outline: Hermione has been thinking about the imminent return of the Dark Lord, so she has been busy packing her bag with all the items required for her survival. Because she has so many different items, it is impossible to list them all here; however she knows that she can formulate the problem even without knowing those (trivial) details. She has $N$ items indexed from $1$ to $N$; each item $x_i$ is associated with a value $c_i$, weight $w_i$ and volume $v_i$. She cannot carry more than $W$ in weight, and the bag can only hold up to $V$ in volume. Items must either be in the backpack or not; i.e. we cannot put half a book in the bag! She needs to maximize the value of the items that she is carrying, because she knows she will not be able to replenish these for a very long time.

How would I formulate this problem using summation notation?

Answer 1

Maximize $\sum c_i$ subject to $\sum w_i\le W$ and $\sum v_i\le V$.

Answer 2

I would say that you take $a\in \{0,1\}^N$ with elements $a_k$, $k=1,\dots,N$ and maximize $$ \begin{cases}\sum_{k=1}^N a_ic_i\to\max_{a\in \{0,1\}^N}\\ \sum_{k=1}^N a_iw_i\le W,\\ \sum_{k=1}^N a_iv_i\le V. \end{cases} $$