1. Solution methodologies for the classical Grant van Dieman Friday 30 th November 2007 Supervisor: Prof. JH van Vuuren Co-supervisor: Mr JN Roux assignment problem

2. Slide 2 Overview  The classical assignment problem  Exact Solution methods  A maximum matching algorithm  Successive shortest path method  Hungarian method  Greedy heuristics  Comparison  Future work

3. Slide 3 The classical assignment problem  Votaw and Orden (1952)  Assumptions  x ij is 1 if assignee i is assigned to task j and 0 otherwise  The assignment problem is NP complete (Lloyd and Witzenhausen (1986))

4. Slide 4 The Weapon Target Assignment Problem  Flood (1957)  V j : priority of eliminating target j.  q ij : is the survival probability of target j if it is engaged by weapon i.  x ij =1 if weapon i engage target j and 0 otherwise

5. Slide 5 Overview  The classical assignment problem  Exact Solution methods  A maximum matching algorithm  Successive shortest path method  Hungarian method  Greedy heuristics  Comparison  Future work

6. Slide 6 A maximum matching algorithm for weighted bipartite graphs (MWM) q ij V 1 = {assignees} V 2 = {tasks} G :

7. Slide 7 A maximum matching algorithm for weighted bipartite graphs (MWM) V 1 = {assignees} V 2 = {tasks} q ij M :

8. Slide 8 Overview  The classical assignment problem  Exact Solution methods  A maximum matching algorithm  Successive shortest path method  Hungarian method  Greedy heuristics  Comparison  Future work

9. Slide 9 Successive shortest path algorithm (SSP)  Minimum cost flow algorithm  Why this algorithm can be used to solve the assignment problem  The value of x ij will be binary

10. Slide 10 Overview  The classical assignment problem  Exact Solution methods  Successive shortest path method  A maximum matching algorithm  Hungarian method  Greedy heuristics  Comparison  Future work

11. Slide 11 Hungarian Method  Kuhn(1955)  Special algorithm for the assignment problem  Construct reduced cost matrix

12. Slide 12 Overview  The classical assignment problem  Exact Solution methods  Successive shortest path method  A maximum matching algorithm  Hungarian method  Greedy heuristics  Comparison  Future work

13. Slide 13 Greedy Heuristics  Greedy RTB  Greedy RBT  Greedy RR  Greedy CLR  Greedy CRL  Greedy CR

14. Slide 14 Overview  The classical assignment problem  Exact Solution methods  Successive shortest path method  A maximum matching algorithm  Hungarian method  Greedy heuristics  Comparison  Future work

15. Slide 15 Comparisons  Benchmark set 1: JE Beasly (Randomly Generated)  3.4 Ghz, 1024 MB ram, Windows XP

16. Slide 16 Comparisons

17. Slide 17 Comparisons

18. Slide 18 Comparisons  Benchmarks set 2: Randomly Generated in Matlab

19. Slide 19 Comparisons

20. Slide 20 Comparisons

21. Slide 21 Future work  Advanced Heuristics and Meta-heuristics  More exact solution methods  Expand algorithms to solve variations of the assignment problem

22. Slide 22 References [1] [2] [3] [4] [5]