Solve the following instance of greedy knapsack problem where n=4, m=10, p = (40, 42, 25,12) and w = (4, 7, 5, 3)

The greedy knapsack problem is a problem in which we are given a set of items, each with a weight and a value, and we need to determine the most valuable combination of items to include in a knapsack of limited weight.

Here, n=4 (number of items), m=10 (maximum weight that the knapsack can hold), p = (40, 42, 25,12) (values of the items), and w = (4, 7, 5, 3) (weights of the items).

Step 1: Calculate the value-to-weight ratio for each item. This is done by dividing the value of each item by its weight.

Item 1: 40/4 = 10 Item 2: 42/7 = 6 Item 3: 25/5 = 5 Item 4: 12/3 = 4

Step 2: Sort the items by their value-to-weight ratio in descending order. The order of items is now 1, 2, 3, 4.

Step 3: Start adding items to the knapsack, starting with the item with the highest value-to-weight ratio. Continue adding items until adding the next item would exceed the maximum weight.

Add item 1: current weight = 4, remaining weight = 10 - 4 = 6 Add item 2: current weight = 4 + 7 = 11, which exceeds the maximum weight. So, we can't add item 2. Add item 3: current weight = 4 + 5 = 9, remaining weight = 10 - 9 = 1 We can't add item 4 because its weight (3) exceeds the remaining weight.

So, the solution to this instance of the greedy knapsack problem is to include items 1 and 3 in the knapsack. The total value of the items in the knapsack is 40 + 25 = 65.