The Knapsack Problem

Abstract

Suppose we are planning a hiking trip; and we are, therefore, interested in filling a knapsack with items that are considered necessary for the trip. There are N different item types that are deemed desirable; these could include bottle of water, apple, orange, sandwich, and so forth. Each item type has a given set of two attributes, namely a weight (or volume) and a value that quantifies the level of importance associated with each unit of that type of item. Since the knapsack has a limited weight (or volume) capacity, the problem of interest is to figure out how to load the knapsack with a combination of units of the specified types of items that yields the greatest total value. What we have just described is called the knapsack problem. A large variety of resource allocation problems can be cast in the framework of a knapsack problem. The general idea is to think of the capacity of the knapsack as the available amount of a resource and the item types as activities to which this resource can be allocated. Two quick examples are the allocation of an advertising budget to the promotions of individual products and the allocation of your effort to the preparation of final exams in different subjects. Formally, suppose we are given the following parameters: