Solving 0-1 knapsack problems based on amoeboid organism algorithm
- 1. Solving 0-1 knapsack problems based on amoeboid organism algorithm Xiaoge Zhang, Shiyan Huang, Yong Hu, Yajuan Zhang, Sankaran Mahadevan, Yong Deng Juan José Miramontes Sandoval Modelos de Sistemas de Software 1
- 2. Tabla de Contenido • 0-1 Knapsack problem • Amoeboid organism • Proposed method ▫ Example with 4 items ▫ Example with 8 items • Experimental results • Conclusiones 2
- 3. 0-1 Knapsack Problem Given a set of items, each with a weight and a value, determine the count of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. 3 K. Pepper. Nakagaki. (2007). Knapsack Taken from: http://en.wikipedia.org/wiki/Knapsack_problem Combinatorial optimization problem
- 4. 0-1 Knapsack Problem Mathematical model: 4
- 5. 0-1 Knapsack Problem This model is widely used in real-life applications: • Capital budgeting problems • Loading problems • Resource allocation • Project selection problems and can be found as a sub problem of other more general models 5
- 6. Amoeboid organism 6 P. Monzon. Alimentación Ameba. Taken from: http://vidadeamebas.blogspot.mx/ • Is a type of cell or organism which has the ability to alter its shape. • Lacking cell wall. • Capture food through movement.
- 7. Proposed method • Recently, it is shown that an amoeboid organism can find the shortest path between two selected points in a labyrinth. A new method using the amoeboid organism model is propose to solve the 0-1 knapsack problem 7 T. Nakagaki. (2001). Tracking the shortest path in a maze by the plasmodium. Effective method to solve optimization problems
- 9. Proposed method Example: Knapsack capacity: W = 6 vj is the value of item j wj is the weight of item j 9 j 1 2 3 4 vj 40 15 20 10 wj 4 2 3 1
- 10. Proposed method 1.- Converting the 0-1 knapsack problem as the longest path problem: 10
- 11. 2.- Transforming the longest path problem into the shortest path problem: 11 Proposed method
- 12. Proposed method 2.- Transforming the longest path problem into the shortest path problem: 12
- 13. 3.- Finding the shortest path based on the amoeboid organism algorithm: Step 1. Remove the edges with conductivity equal to zero. Step 2. Calculate the pressure of each node using each node’s current conductivity and length. Step 3. Use the pressure of each node obtained from step 2 to calculate each node’s conductivity. Step 4. Judge whether each edge’s conductivity is 1, if not, go to step 5; otherwise go to step 7. Step 5. According to the current flux and conductivity, calculate the flux and conductivity next time. Step 6. Return to step 1. Step 7. Get the result and the algorithm is over. 13 Proposed method
- 14. • According to the network converting algorithm shown previously, there are 퐖 + ퟏ ∗ 풏 + ퟐ items in the converted network. • For the amoeboid organism algorithm, regardless of the algorithm’s outer iterations, its main time is spent on solving the linear equations and its complexity is 푶(풏ퟑ) Then, the complexity of the of the proposed method is: 퐎 ( 푾 + ퟏ ∗ 풏 + ퟐ)ퟑ = 푶(풏ퟑ) 14 Complexity of the proposed method
- 15. Proposed method Example with 8 items: Knapsack capacity: W = 8 vj is the value of item j wj is the weight of item j 15 j 1 2 3 4 5 6 7 8 vj 83 14 54 79 72 52 48 62 wj 3 2 3 2 1 2 2 3
- 16. 1.- Converting the 0-1 knapsack problem as the longest path problem: 16
- 17. 17 2.- Transforming the longest path problem into the shortest path problem:
- 18. 18 3.- Applying the amoeboid organism algorithm:
- 19. Experimental results 19 Result of knapsack problem with 8 Items: J 1 2 3 4 5 6 7 8 vj 83 14 54 79 72 52 48 62 wj 3 2 3 2 1 2 2 3 Six test problems with different dimensions are used to study the performance of the proposed method:
- 20. Conclusions 20 • Based on amoeboid organism algorithm and the network converting algorithm, a new method is proposed to solve classical 0-1 knapsack problems. • Using benchmark problems to test the amoeboid organism algorithm, the computational results demonstrate the efficiency of the presented approach.
- 21. References • Zhang, X., Huang, S., Hu, Y., Zhang, Y., Mahadevan, S., & Deng, Y. (2013). Solving 0-1 knapsack problems based on amoeboid organism algorithm. Applied Mathematics and Computation, 219(19), 9959–9970. doi:10.1016/j.amc.2013.04.023 21