In the Fractional Knapsack problem, if there are 3 items with weights [10, 20, 30] and values [60, 100, 120], and the capacity of the knapsack is 50, what is the maximum value that can be obtained?

Homework Help: Questions and Answers: In the Fractional Knapsack problem, if there are 3 items with weights [10, 20, 30] and values [60, 100, 120], and the capacity of the knapsack is 50, what is the maximum value that can be obtained?

a) 240 b) 220 c) 180 d) 200

Answer:

The Fractional Knapsack problem allows us to take fractions of items, meaning that we can take any portion of an item to maximize the total value within a given capacity. To solve the problem, follow these steps:

Value-to-Weight Ratio for Each Item

For each item, we calculate the value-to-weight ratio, which helps in deciding which item (or fraction of item) to take first to maximize the total value.

Item 1: weight = 10, value = 60 Value-to-weight ratio = 60/10=6

Item 2: weight = 20, value = 100 Value-to-weight ratio = 100/20=5

Item 3: weight = 30, value = 120 Value-to-weight ratio = 120/30=4

Sort Items by Value-to-Weight Ratio

Sort the items in descending order of value-to-weight ratio:

Item 1 (ratio = 6)

Item 2 (ratio = 5)

Item 3 (ratio = 4)

Items in Order of Highest Ratio

Now, we fill the knapsack based on the sorted order of items, aiming to maximize the value without exceeding the capacity of 50 units.

Item 1: Take the entire item, as its weight (10) is less than the remaining capacity (50). Remaining capacity = 50−10 = 40 Total value = 60

Item 2: Take the entire item, as its weight (20) is less than the remaining capacity (40). Remaining capacity = 40−20 = 20 Total value = 60+100 = 160

Item 3: Only 20 units of capacity remain, so we take 20/30 fraction of Item 3. Value from Item 3 = (20/30)×120 = 80 Total value = 160+80 = 240