4.5 0/1 Knapsack - Two Methods - Dynamic Programming

Step-by-Step Tutorial: Solving the 0/1 Knapsack Problem using Dynamic Programming

Overview:

In this tutorial, we will learn how to solve the 0/1 Knapsack Problem using Dynamic Programming. The problem involves maximizing the total profit by selecting objects with specific weights to fit into a knapsack of limited capacity.

Steps:

1. Understanding the Problem:

   • The problem involves selecting objects with profits and weights to fit into a knapsack of limited capacity.
   • The objective is to maximize the total profit while ensuring the total weight of the selected objects does not exceed the knapsack's capacity.

2. Approach of Dynamic Programming:

   • Dynamic Programming involves solving optimization problems by making a sequence of decisions.
   • The decisions involve whether to include or exclude each object in the knapsack.
   • The approach is to try all possible solutions and select the best one.

3. Tabulation Method:

   • Create a table with columns representing the knapsack's capacity and rows representing objects.
   • Fill the table using a formula to calculate the maximum profit that can be achieved.
   • Start by considering each object one by one and calculating the profits for different capacities of the knapsack.

4. Sets Method:

   • Generate sets of ordered pairs representing profit and weight combinations for each object.
   • Merge the sets to create all possible combinations of objects that can be included in the knapsack.
   • Apply the dominance rule to eliminate invalid combinations and select the best set of objects with maximum profit.

5. Sequence of Decisions:

   • Identify the maximum profit in the final set obtained from the Sets Method.
   • Trace back the decisions made to include or exclude objects by subtracting their profits and weights from the maximum profit achieved.
   • Determine the objects to be included in the knapsack based on the sequence of decisions made.

6. Final Solution:

   • Based on the sequence of decisions, determine which objects are included in the knapsack to maximize the total profit.
   • Write down the values of X1, X2, X3, X4 representing whether each object is included (1) or not included (0).
   • Calculate the total profit obtained by including the selected objects in the knapsack.

7. Verification:

   • Verify the solution obtained using both the Tabulation Method and the Sets Method to ensure consistency and accuracy.
   • Compare the results obtained from both methods to validate the solution for the 0/1 Knapsack Problem.

By following these steps, you can effectively solve the 0/1 Knapsack Problem using Dynamic Programming and choose the best approach based on the complexity of the problem and the desired outcome.