This is a Python program to solve the fractional knapsack problem using greedy algorithm.

In the fractional knapsack problem, we are given a set of n items. Each item i has a value v(i) and a weight w(i) where 0 <= i < n. We are given a maximum weight W. The problem is to find how much of each item we should take such that the total weight does not exceed W and the total value is maximized. Thus, we want to find f such that sum of v(i)f(i) over all i is maximized, w(i)f(i) <= W for all i and 0 <= f(i) <= 1 for all i.

1. The function fractional_knapsack is defined.

2. It takes three arguments: two lists value and weight; and a number capacity.

3. It returns (max_value, fractions) where max_value is the maximum value of items with total weight not more than capacity.

4. fractions is a list where fractions[i] is the fraction that should be taken of item i, where 0 <= i < total number of items.

5. The function works by choosing an item from the remaining items that has the maximum value to weight ratio.

6. If the knapsack can include the entire weight of the item, then the full amount of the item is added to the knapsack.

7. If not, then only a fraction of this item is added such that the knapsack becomes full.

8. The above three steps are repeated until the knapsack becomes full, i.e. the total weight reaches the maximum weight.

Here is the source code of a Python program to solve the fractional knapsack problem using greedy algorithm. The program output is shown below.

def fractional_knapsack(value, weight, capacity): """Return maximum value of items and their fractional amounts. (max_value, fractions) is returned where max_value is the maximum value of items with total weight not more than capacity. fractions is a list where fractions[i] is the fraction that should be taken of item i, where 0 <= i < total number of items. value[i] is the value of item i and weight[i] is the weight of item i for 0 <= i < n where n is the number of items. capacity is the maximum weight. """ # index = [0, 1, 2, ..., n - 1] for n items index = list(range(len(value))) # contains ratios of values to weight ratio = [v/w for v, w in zip(value, weight)] # index is sorted according to value-to-weight ratio in decreasing order index.sort(key=lambda i: ratio[i], reverse=True) max_value = 0 fractions = [0]*len(value) for i in index: if weight[i] <= capacity: fractions[i] = 1 max_value += value[i] capacity -= weight[i] else: fractions[i] = capacity/weight[i] max_value += value[i]*capacity/weight[i] break return max_value, fractions n = int(input('Enter number of items: ')) value = input('Enter the values of the {} item(s) in order: ' .format(n)).split() value = [int(v) for v in value] weight = input('Enter the positive weights of the {} item(s) in order: ' .format(n)).split() weight = [int(w) for w in weight] capacity = int(input('Enter maximum weight: ')) max_value, fractions = fractional_knapsack(value, weight, capacity) print('The maximum value of items that can be carried:', max_value) print('The fractions in which the items should be taken:', fractions)

1. The user is prompted to enter the number of items n.

2. The user is then asked to enter n values and n weights.

3. The function fractional_knapsack is called to get the maxmimum value and the list of fractions.

4. The result is then displayed.

Case 1: Enter number of items: 3 Enter the values of the 3 item(s) in order: 60 100 120 Enter the positive weights of the 3 item(s) in order: 10 20 30 Enter maximum weight: 50 The maximum value of items that can be carried: 240.0 The fractions in which the items should be taken: [1, 1, 0.6666666666666666] Case 2: Enter number of items: 5 Enter the values of the 5 item(s) in order: 3 5 1 2 4 Enter the positive weights of the 5 item(s) in order: 40 50 20 10 30 Enter maximum weight: 75 The maximum value of items that can be carried: 9.5 The fractions in which the items should be taken: [0, 0.7, 0, 1, 1] Case 3: Enter number of items: 1 Enter the values of the 1 item(s) in order: 5 Enter the positive weights of the 1 item(s) in order: 10 Enter maximum weight: 5 The maximum value of items that can be carried: 2.5 The fractions in which the items should be taken: [0.5]

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