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Aiying Rong 1 , Jose Rui Figueira 2 , Margarida Vaz Pato 3

A Two State Reduction Based Dynamic Programming Algorithm for the Bi-Objective 0-1 Knapsack Problem. Aiying Rong 1 , Jose Rui Figueira 2 , Margarida Vaz Pato 3 1 Cemapre (Center of Applied Mathematics and Economics) ISEG – Technical University of Lisbon, Lisbon, Portugal

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Aiying Rong 1 , Jose Rui Figueira 2 , Margarida Vaz Pato 3

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  1. A Two State Reduction Based Dynamic Programming Algorithm for the Bi-Objective 0-1 Knapsack Problem Aiying Rong 1, Jose Rui Figueira 2, Margarida Vaz Pato 3 1 Cemapre (Center of Applied Mathematics and Economics) ISEG – Technical University of Lisbon, Lisbon, Portugal 2 INPL, Ecole des Mines de Nancy Laboratoire LORIA, Nancy, France 3 Centrode Investigação Operacional FC - University of Lisbon ISEG - Technical University of Lisbon , Lisbon, Portugal 21st MCDM June 13-17th, 2011

  2. Outline • Introduction • Problem statement • Motivation and existing research • Two reduction based DP algorithm • Backward reduced state DP space • Sparse node • Numerical experiment • Typical test instances • Comparison with the state-art of the DP algorithm 21st MCDM June 13-17th, 2011

  3. Introduction Problem Statement & Given n items and r profit objectives & each item j with weight wjand the kthe profit objective j= 1,.., n, k = 1,…,r and & a knapsack with capacity W & selecting a subset of items with total weight not exceeding W and total profit objectives maximized in the Pareto sense 21st MCDM June 13-17th, 2011

  4. Introduction • Existing Research • Branch and bound algorithm (BB) • Dynamic programming (DP) with labelling algorithms (LDP) • DP with several dominance relations (DDP) • Exact algorithm by exploring the development of multi-objective linear programming • DDP  LDP BB • DP is one of the best approaches 21st MCDM June 13-17th, 2011

  5. Introduction • Motivation • LDP Applying a forward DP space but computation effort in the last stage is too heavy • DDP ( DP+BB): time consuming dominance checking • Hard instances difficult to solve • Harder instance: confilicting objectives uncorrelated with weight coefficients • Hardest instance: conflicting objectives positively correlated with weight coefficients • The number of non-dominace objective vectors increases singificatntly with the problem size 21st MCDM June 13-17th, 2011

  6. Two reduction based on DP(TDP) Two reduction techniques Backward reduced state DP space (primary) Sparse nodes (secondary) Nodes with distinct objective vectors The same objective vectors between two consecutitve sparse nodes. Only one node needs to be calcalculated. Illustrative example “max” (7x1 + 9 x2 + 3x3 + 7x4 +6x5, 2x1 + 2x2 + 10x3 + 6x4 +9x5) subject to: 3x1 + 2 x2 + 2x3+ 4x4 +3x5 9, xj{0,1}, j = 1,…,5. 21st MCDM June 13-17th, 2011

  7. TDP Attractive to multi-objective case But not to single objective case O(nW1) time and O(nW1) space Non-increasing order node pattern Singificant reduction of Computational effort and memory requirement Backward reduced state DP space 21st MCDM June 13-17th, 2011

  8. TDP 21st MCDM June 13-17th, 2011

  9. Numerical results • Four types of instances • Uncorrelated instances (Type A) • Uconflicting instances :Postively corelated profit objecives correlated with weitht coefficients (Type B) • Conflicting instances (I): conflicting profit objectives uncorrelated with weight coefficients (Type C) • Conflicting instances (II): conflicting objectives correlated with weight coefficients (Type D) • 30 instances for each problem size and each type • Purpose • performance of the TDP • Contribution of two reduction techniques • Computional facility • A 2.49 GHz Pentium PC with 2.98 GB RAM under windows XP operating systems • C++ in the Microsoft visual studio 2003 environment 21st MCDM June 13-17th, 2011

  10. Numerical results CPU Comparison results of TDP and RDP with DDP 21st MCDM June 13-17th, 2011

  11. Numerical results CPU Comparison results of TDP and RDP with DDP 21st MCDM June 13-17th, 2011

  12. Numerical results CPU Comparison results of TDP and RDP with DDP 21st MCDM June 13-17th, 2011

  13. Numerical results Overall performance indicators (type A) 21st MCDM June 13-17th, 2011

  14. Numerical results Overall performance indicators (type C) 21st MCDM June 13-17th, 2011

  15. Numerical results Overall performance indicators (type D) 21st MCDM June 13-17th, 2011

  16. Numerical results Relation between Avecotors and CPU time 21st MCDM June 13-17th, 2011

  17. Conclusion • Two reduction based DP algorithm • The backward reduced DP space partiularly effective for the multi-objective case • Reduction techniques defined for any number of profit objective, more effective than the dominance relations in the DDP algorithm 21st MCDM June 13-17th, 2011

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