Given n items, each with a profit and a weight, given a knapsack of
capacity c, the goal is to find a subset of items which fits inside c
and maximizes the total profit.
The knapsack problem can easily be extended from 1 to d dimensions.
As an example, this can be useful to constrain the maximum number of
items inside the knapsack.
Without loss of generality, profits and weights are assumed to be positive.

From a mathematical point of view, the multi-dimensional knapsack problem
can be modeled by d linear constraints:

ForEach(j:1..d)(Sum(i:1..n)(weight_ij * item_i) <= c_j
where item_i is a 0-1 integer variable.

Then the goal is to maximize:

Sum(i:1..n)(profit_i * item_i).

There are several ways to solve knapsack problems.