Knapsack

Years and Authors of Summarized Original Work

2000; Ibarra, Kim

Problem Definition

For a given set of items N = {1, …, n} with nonnegative integer weights w _j and profits p _j , j = 1, …, n, and a knapsack of capacity c, the knapsack problem (KP) is to select a subset of the items such that the total profit of the selected items is maximized and the corresponding total weight does not exceed the knapsack capacity c.

Alternatively, a knapsack problem can be formulated as a solution of the following linear integer programming formulation:

$$\displaystyle{ \begin{array}{*{20}c} \left (KP\right )&\text{maximize}\sum \limits _{j=1}^{n}p_{i}x_{j}\\ \end{array} }$$

(1)

$$\displaystyle{ \text{subject to}\,\sum \limits _{j=1}^{n}w_{ j}x_{j} \leq c, }$$

(2)

$$\displaystyle{ x_{j} \in \left (0,1\right ),\;j = 1,\ldots,n. }$$

(3)

The knapsack problem is the simplest nontrivial integer programming model having binary variables, only a single constraint, and only positive coefficients. A large...