modeling

The Knapsack Problem

In this exercise, you'll be asked to implement several variants of the famous Knapsack problem in knapsack.py.

The most popular variant of this problem is the $0-1$ knapsack, where one needs to decide which items to put into a knapsack/bag. These items both have a weight and a value, and thus the objective is to maximize the total value we bring, keeping in mind that the knapsack's capacity cannot be exceeded by the total weight of chosen items.

$$ \begin{align*} \max_x & \sum_{i \in \mathcal{I}} x_i v_i \\ \text{s.t.} & \sum_{i \in \mathcal{I}} x_i w_i \leq C\\ & x_i \in \lbrace 0,1 \rbrace, \forall i \in \mathcal{I} \end{align*} $$

Figure 1: Skippy undecided between taking a gold bar, a pizza slice, and the Z1 motor-driven mechanical computer.

Sometimes the key to solving a real-world problem is to identify the well-known classic problem that represents them.
Here are some examples where knapsack could be applied:

• Loading cargo in airplanes given their cost and weight
• Choosing which ads to display on a webpage, given revenue and size constraints.
• Maximizing portfolio revenue given risk and diversification constraints

Exercise 1: Knapsack problem with fractional items

Let us start with a simplified version of the classical knapsack, where instead of choosing which items to pick, we must choose which amount to pick. Mathematically, the variable bounds go from $x_i \in \lbrace 0,1 \rbrace$ to $0 \leq x_i \leq 1$.

Your task: Formulate the problem described above with continuous variables in knapsack.py, method linear_knapsack().

Hint 1

The variable bounds can be set during variable creation with the options lb and ub, for lower bound and upper bound, respectively.

Exercise 2: 0-1 Knapsack problem

In many scenarios, we are more interested in the binary variant of the knapsack problem.

Your task: Enforce integrality on the variables, in the method binary_knapsack().

Hint 1

You can set the variable type during variable creation with the vtype option.

Exercise 3: Allow many items

Your task: Allow the possibility for choosing multiple copies of the same item, in the method integer_knapsack().

Hint 1

Instead of the variables being binary, they can just be integer.

Exercise 4: Max n items in the knapsack

Your task: Force the solution from integer knapsack to have at most n items, in the method limited_knapsack().