指導教授 : 楊能舒 教授 學生 : 陳泓翔

1. 以分支界限法為基礎的啟發式方法求解二次指派問題以分支界限法為基礎的啟發式方法求解二次指派問題 A heuristic method based on branch and bound algorithmfor solving quadratic assignment problems 指導教授:楊能舒 教授 學生:陳泓翔

2. Reporting process 三 Research Methods 一 Introduction 二 Literature review 四 五 Scope of  Research Conclusion & Timetable Background Spreadsheet Conclusion Motivation Branch And Bound Literature Table Research Plan Research Objective Heuristic Method Timetable Research Process Solution Procedure

3. 1.Introduction-Background Department 1 Location One department Assign to One location Department 3 Department 4 Department 2 Matrix B:the interaction among the department Matrix U: Assigned matrix Matrix A:the distance between location

4. 1.Introduction-Background If the flow matrix is symmetrical Total cost(2134) 1 17+27+40 2 3 + 4 18+34 3 + 1 4 25 3 4 51 But there exist 4!=24 different layouts if we don’t know optimal solution

5. 1.Introduction-Motivation Spreadsheets (Rasmus2007) frameworkmethod Low complexity QAP Use LAPto solve (francis and white 1974) Bound Value generated trouble! Branch and Bound High complexity Better ? This research try to use Genetic Algorithm

6. 1.Introduction-Research objective QAP Branchand Bound Genetic Algorithm search: Expect to get better Bound value Use LAP to Calculate Bound Value generated trouble! A heuristic method Use A heuristic method To reduce the complexity of branch and bound for solving quadratic assignment problems

7. 1.Introduction-Research objective QAP Low complexity High complexity Spreadsheets Spreadsheets Good solution quality and speed Can not be solved Use A heuristic method compared with Spreadsheets , expect to solve Spreadsheets’s defect

8. 1.Introduction-Research Process

9. 2.Literature review-Literature table

10. 2.Literature review-Literature table

11. 3. Research Methods-Spreadsheets Department 1 Location One department Assign to One location Department 3 Department 4 Department 2 Matrix B:the interaction among the department Matrix U: Assigned matrix Matrix A:the distance between location

12. 3. Research Methods-Spreadsheets R. Rasmussen(2007)提出spreadsheets的方法 A: Distance matrix B: Interaction flow matrix U: Assignment matrix

13. 3. Research Methods-Spreadsheets B U Min S = TRACE 1 0 0 0 T 0 1 0 0 0 0 1 0 0 0 0 1 A U = 51 0784 7070 8705 4050 1000 0001 0100 0010 0784 7070 8705 4050 1000 0001 0100 0010

14. 3. Research Methods-Spreadsheets Can be solved by programming solver and obtain the minimum cost solution of this equation by Spreadsheets

15. 3. Research Methods-Branch&Bound (1．．．．．) (2．．．．．) (3．．．．．) (5．．．．．) (6．．．．．) (4．．．．．) (24．．．．) (25．．．．) (21．．．．) (23．．．．) (26．．．．)

16. 3. Research Methods-Branch&Bound If (2 1 ．．．．) are assigned AssignedUnassigned 4/? 2/1 3/? 2 8 2 2 2 2 2 8 5/? 6/? 1/2 2

17. 3. Research Methods-Branch&Bound B21= d(a(2)=1,a(1)=2)+d(a(1)=2,a(2)=1) Between already assigned departments + d(a(2)=1,a(3)=?)+d(d(a(2)=1,a(4)=?)+ … + d(a(1)=2,a(3)=?)+d(d(a(1)=2,a(4)=?)+ … Between already assigned departments and not yet assigned departments + d(a(3)=?,a(4)=?)+d(d(a(3)=?,a(5)=?)+ … Between not yet assigned departments The number of all the Wxd arcs symmetric and non-symmetric are 30

18. 3. Research Methods-Branch&Bound w21xd(a(2)=1,a(1)=2) = w12xd(a(1)=2,a(2)=1) w23xd(a(2)=1,a(3)=?) = w32xd(a(3)=?,a(2)=1) w34xd(a(3)=?,a(4)=?) = w43xd(a(4)=?,a(3)=?) Save the calculations by half But in real world, distance may be symmetric, flow isusually not What if the Flow and Distance Matrix are symmetric

19. 3. Research Methods-Branch&Bound Let partial assignment denote the locations of Departments 1,2,3．．．q, where qM

20. 3. Research Methods-Branch&Bound Assigned departments to other assigned departments Yet to be assigned departments to assigned departments The interaction cost of unassigned departments

21. 3. Research Methods-Branch&Bound If (2 1 ．．．．) assigned AssignedUnassigned 2/1 3/? 4/? 1/2 5/? 6/?

22. 3. Research Methods-Branch&Bound Assigned departments to other assigned departments Yet to be assigned departments to assigned departments The interaction cost of unassigned departments What we know The flow and distance between those departments already assigned to specific locations

23. 3. Research Methods-Branch&Bound Assigned departments to other assigned departments Yet to be assigned departments to assigned departments The interaction cost of unassigned departments What we do not know, But we can Guess in a logical sense The flow and distance between those departments not yet assigned to specific locations

24. 3. Research Methods-Branch&Bound Assigned departments to other assigned departments Yet to be assigned departments to assigned departments The interaction cost of unassigned departments

25. 3. Research Methods-Branch&Bound What to do with those un assigned departments assign these departments to every possible location An Optimal Solution can be found by LAP

26. 3. Research Methods-Branch&Bound Assign Department 3 to Location 3 w34 4 4 d34 w35 5 5 d35 w36 6 6 d36 3/3 Department Location

27. 3. Research Methods-Branch&Bound

28. 3. Research Methods-Branch&Bound The cheapest way is to use the shortest distances for the highest flow volumes WxD= [w41 w45 w46] x d34 d35 d36

29. 3. Research Methods-Branch&Bound Calculated for each value of the G matrix to find the best arrangement unassigned 3 4 5 6 G =3g33 g34 g35 g36 4g43g44 g45 g46 5 g53 g54 g55 g56 6 g63 g64 g65 g66 + w*d (assigned) = Bound value of Node

30. 3. Research Methods-Branch&Bound Backtracking :Search Procedure B=30 B=15 B=32 B=18 B=25 B=28 B=35 B=20 B=38 B=33 B=39 (1．．．．．) (2．．．．．) (3．．．．．) (5．．．．．) (6．．．．．) (4．．．．．) (21．．．．) (24．．．．) (25．．．．) (23．．．．) (26．．．．)

31. 3. Research Methods-Branch&Bound Live Search List (B1=30, B2=15, B3=32, B4=18, B5=25, B6=28) (B1=31, B21=35, B23=20, B24=33, B25=38, B26=39, B3=32, B4=18, B5=25, B5=28)

32. 3. Research Methods- Heuristic method B=30 B=15 B=32 B=18 B=25 B=28 B=35 B=20 B=38 B=33 B=39 (1．．．．．) (2．．．．．) (3．．．．．) (5．．．．．) (6．．．．．) (4．．．．．) (21．．．．) (24．．．．) (25．．．．) (23．．．．) (26．．．．)

33. 3. Research Methods-Heuristic method Why Boundvalue have problem? WxD= [w41 w45 w46] x d34 d35 d36 3 4 5 6 G =3g33 g34 g35 g36 4g43g44 g45 g46 5 g53 g54 g55 g56 6 g63 g64 g65 g66

34. 3. Research Methods- Heuristic method

35. 3. Research Methods- Genetic Algorithm So we do not use LAP find the solutions, use GA based on branch and bound method to find a better solutions 1.[coding] Department 1（1）、 Department 2（2）、 Department 3（3）、 、、、 Department 6（6） 2.[Initialization] Ex: 213456123456132546345612 3. [fitness function] Ex:f(213456) = 30

36. 3. Research Methods- Genetic Algorithm 4. [selection]selected for next generation 5. [crossover] IfS1=241356S2=214563 thenS1=2143563 S2=214356 6. [mutation] Before mutationafter mutation241356261354 7. Repeat 4. 5. 6 until end

37. 3. Research Methods- Heuristic method What to do with those unassigned departments assign these departments to every possible location So we use GA Try to found better Boundvalue

38. 3. Research Methods- Heuristic method (1．．．．．) (2．．．．．) (3．．．．．) (5．．．．．) (6．．．．．) (4．．．．．) (24．．．．) (25．．．．) (21．．．．) (23．．．．) (26．．．．)

39. 3. Research Methods-Method Procedure QAP Branch&Bound Use LAP Bound value has problem! Use GA Good Solution Good Solution Different!

40. 4.Scope of Research-Research plan Compared following algorithms to explore the feasibility of the new heuristic method 1.Branch and Bound 2.Spreadsheets 3. Heuristic method

41. 4.Scope of Research-Research plan According to the problems characteristic comparison methods

42. 5. Conclusion This study proposes a heuristic method based on branch and bound algorithmfor solving quadratic assignment problems, aims to improve the LAP for solving quadratic assignment problems’s defect . In this research, we will compared three methods with the different kind of problems, and look forward to providing a solving method for solving quadratic assignment problem’s people.

43. 5. Timetable

44. Thanks for your listening