Feuerman and Weiss proposed a system in which students are given a heterogeneous test with a total of 125 possible points. However, on tests with a heterogeneous distribution of point values, it is more difficult to provide choices. For example, if an exam contains 12 questions each worth 10 points, the test-taker need only answer 10 questions to achieve a maximum possible score of 100 points. For small examples, it is a fairly simple process to provide the test-takers with such a choice. One early application of knapsack algorithms was in the construction and scoring of tests in which the test-takers have a choice as to which questions they answer. Knapsack problems appear in real-world decision-making processes in a wide variety of fields, such as finding the least wasteful way to cut raw materials, selection of investments and portfolios, selection of assets for asset-backed securitization, and generating keys for the Merkle–Hellman and other knapsack cryptosystems. The name "knapsack problem" dates back to the early works of the mathematician Tobias Dantzig (1884–1956), and refers to the commonplace problem of packing the most valuable or useful items without overloading the luggage. The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. The problem often arises in resource allocation where the decision-makers have to choose from a set of non-divisible projects or tasks under a fixed budget or time constraint, respectively. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items.
Given a set of items, each with a weight and a value, determine which items to include in the collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. The knapsack problem is the following problem in combinatorial optimization: (Solution: if any number of each box is available, then three yellow boxes and three grey boxes if only the shown boxes are available, then all except for the green box.) Example of a one-dimensional (constraint) knapsack problem: which boxes should be chosen to maximize the amount of money while still keeping the overall weight under or equal to 15 kg? A multiple constrained problem could consider both the weight and volume of the boxes.
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