Q1. The Knapsack problem is a problem in combinatorial optimization. This problem also arises in resource allocation whe

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Q1 The Knapsack Problem Is A Problem In Combinatorial Optimization This Problem Also Arises In Resource Allocation Whe 1 (74.86 KiB) Viewed 34 times

Q1. The Knapsack problem is a problem in combinatorial optimization. This problem also 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. As such, this problem can be applied in many different contexts. Given a set of items, each with a weight and a value, we want to determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. 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. What makes Branch-and-Bound (B&B) algorithm more efficient than Backtracking in this context is that instead of traversing the tree in a predetermined order (depth-first, breadth-first, etc.), we traverse it based on the optimization criteria for the problem. Problem: Let n items be given, where each item has a weight and a profit. The weights and profits are positive integers. Furthermore, let a positive integer W be given. Determine a set of items with maximum total profit, under the constraint that the sum of their weights cannot exceed W. Inputs: Positive integers n and W, arrays of positive integers w and p, each indexed from 1 to n, and each of which is sorted in non-increasing order according to the values of p/w. Outputs: An integer maxprofit that is the sum of the profits of an optimal set. Write a Java program using the Best-First Search with Branch-and-Bound Pruning algorithm for the 0-1 Knapsack problem and demonstrate its correctness with some input values. Answer: (please write your answer here, add required space if needed)