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In theoretical computer science, the continuous knapsack problem (also known as the fractional knapsack problem) is an algorithmic problem in combinatorial optimization in which the goal is to fill a container (the "knapsack") with fractional amounts of different materials chosen to maximize the value of the selected materials.
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, [3] selection of investments and portfolios, [4] selection of assets for asset-backed securitization, [5] and generating keys for the Merkle–Hellman [6] and other knapsack cryptosystems.
The bin packing problem can also be seen as a special case of the cutting stock problem. When the number of bins is restricted to 1 and each item is characterized by both a volume and a value, the problem of maximizing the value of items that can fit in the bin is known as the knapsack problem.
The knapsack problem is one of the most studied problems in combinatorial optimization, with many real-life applications. For this reason, many special cases and generalizations have been examined. For this reason, many special cases and generalizations have been examined.
Note: consider In the 2-weighted knapsack problem, where each item has two weights and a value, and the goal is to maximize the value such that the sum of squares of the total weights is at most the knapsack capacity: (,) + (,). We could solve it using a similar DP, where each state is (current weight 1, current weight 2, value).
The variant where variables are required to be 0 or 1, called zero-one linear programming, and several other variants are also NP-complete [2] [3]: MP1 Some problems related to Job-shop scheduling; Knapsack problem, quadratic knapsack problem, and several variants [2] [3]: MP9 Some problems related to Multiprocessor scheduling
Best Buy CEO Corie Barry said Tuesday that personal electronics could become more expensive if President-elect Donald Trump carries out his threat to slap new tariffs on foreign goods, as large ...
This method solves the cutting-stock problem by starting with just a few patterns. It generates additional patterns when they are needed. For the one-dimensional case, the new patterns are introduced by solving an auxiliary optimization problem called the knapsack problem, using dual variable information from the linear program.