Knapsack-Problem

Solution of knapsack problem using dynamic programming

Purpose

To get as much value into the knapsack as possible given the weight constraint of the knapsack.

Input

• Maximum weight (capacity of knapsack)
• Item Count
• Item Value
• Item Weight

Output

• Maximum Value
• Chosen Items

Solution Approach

• Evaluate the values of the items iteratively.
• For example, put the first item and select second item. Then evaluate first and second item. Make the most appropriate choice according to the value and weight of the items. And then evaluate the all item.
• In order to get rid of the recalculation, the calculation for the items are kept on a table at each step.
• After the items have been evaluated, the value of V[ItemCount,MaximumWeight] shows the maximum value we can get to the knapsack.

Example

• Item Count = 4
• Max Weight = 5

Item	1	2	3	4
Value	100	20	60	40
Weight	3	2	4	1

Value Matrix

V[i,w]	w=0	1	2	3	4	5
i=0	0	0	0	0	0	0
1	0	0	0	100	100	100
2	0	0	20	100	100	120
3	0	0	20	100	100	120
4	0	40	40	100	140	140

Maximum value we can put the knapsack is V[4,5] = 140

Chosen items are 1 and 4

How the items are selected can be seen in the code