1. 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 algorithm for solving quadratic assignment problems
指導教授:楊能舒 教授 學生:陳泓翔

2. Reporting process 一 Introduction 二 Literature review 三 Research
Methods 四 五
Scope
of 
Research
Conclusion
&
Timetable
Background
Spreadsheet
Conclusion
Motivation
Literature
Table
Branch
And
Bound
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 A:the distance between location
Matrix B:the interaction among the department
Matrix U:
Assigned matrix

4. 1.Introduction-Background
If the flow matrix is symmetrical
Total cost(2134)= 𝑗≠𝑖 𝑤 𝑖𝑗 𝑑 𝑎 𝑖 , 𝑎 𝑗
= 1×7+2×7+4×
1×8+3× 2×
= 51
But there exist 4!=24 different layouts if we don’t know optimal solution

5. 1.Introduction-Motivation
Spreadsheets
(Rasmus 2007)
framework method
Low complexity
QAP
Use LAP to 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
Branch and Bound
Genetic Algorithm
search: Expect to get better
Bound value
A heuristic method
Use LAP to Calculate Bound Value generated trouble!
Use A heuristic method
To reduce the complexity of branch and bound for solving quadratic assignment problems

7. 1.Introduction-Research objective
QAP
High complexity
Low 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 tableNO 作者 年代 方法 結果
和本研究相關性 1
Rasmus Rasmussen
2007
Use EXCEL to solve QAP:Input distance matrix and flow matrix, and develop a set of formulas to solve QAP easier
When solving the Low complexity QAP, have good solution, but can not solve High complexity problems
Provided the Motivation 2
Francis,
White
1974
Use Branch and Bound to solve QAP
Have good solution, but Bound value have the problem
Objective :Improve Branch and Bound 3
Y. Yuan,
S. Omatu
2000
Use GA With Local Search Policy to solve QAP
Compared with genetic algorithms, the result is better than genetic algorithms
genetic algorithms for solving QAP has good results, and can be improved 4
Yong zhong Wu,
Ping Ji
2008
Use GA With New Replacement Strategyto solve QAP
genetic algorithms for solving QAP has good results, and can be improved

10. 2.Literature review-Literature tableNO 作者 年代 方法 結果
和本研究相關性 5
G.Askin,
R.Standridge
1999
QAP definition and introduce a variety of different solutions
These methods are used to solve the QAP can be obtained a good solution
Understand the definition of the QAP and choose the branch and bound method to do for the study of framework 6
Peter Hahn, Thomas Grant,
Nat Hall
1998
Use hungarian method based on branch and bound to solve QAP
Has better solutions
Branch and bound to solve QAP can be improved 7
Hsin-Fu Chen
2004
proposed two methods to speed up the solving efficiency of branch and bound
Understanding branch and bound’s framework and how to improve the branch and bound

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

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 T A U = 51 ×
× 1 2 × × ×

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．．．．．)
(4．．．．．)
(5．．．．．)
(6．．．．．)
(21．．．．)
(23．．．．)
(24．．．．)
(25．．．．)
(26．．．．)

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

17. 3. Research Methods-Branch&Bound
B21= W 21 ×d(a(2)=1,a(1)=2)+ W 12 × d(a(1)=2,a(2)=1) Between already assigned departments + W 23 × d(a(2)=1,a(3)=?)+ W 24 × d(d(a(2)=1,a(4)=?)+ … + W 13 × d(a(1)=2,a(3)=?)+ W 14 × d(d(a(1)=2,a(4)=?)+ … Between already assigned departments and not yet assigned departments + W 34 × d(a(3)=?,a(4)=?)+ W 35 × 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
What if the Flow and Distance Matrix are symmetric
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 is usually not

19. 3. Research Methods-Branch&Bound
Let partial assignment 𝑎 𝑞 = 𝑎 𝑞1 , 𝑎 𝑞2 , 𝑎 𝑞3 ,．．．, 𝑎 𝑞𝑞
denote the locations of Departments 1,2,3．．．q,
where q≤M

20. 3. Research Methods-Branch&Bound
TC 𝑎 𝑞 = 𝑖=1 𝑞 𝐶 𝑖 𝑎 𝑖 + 𝑖=1 𝑞−1 𝑢 𝑚 𝑖≤𝑗≤𝑞 𝑤 𝑖𝑗 𝑑 𝑎 𝑖 , 𝑎 𝑗
+ 𝑗=𝑞+1 𝑀 𝐶 𝑗 𝑎 𝑗 + 𝑖=1 𝑞 𝑤 𝑖𝑗 𝑑 𝑎 𝑖 , 𝑎 𝑗
+ 𝑖=𝑞+1 𝑀−1 𝑗>𝑖 𝑤 𝑖𝑗 𝑑 𝑎 𝑖 , 𝑎 𝑗
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 we calculate the bound value of B 21 by following steps Assigned Unassigned
2/1
3/?
4/?
1/2
5/?
6/?

22. 3. Research Methods-Branch&Bound
TC 𝑎 𝑞 = 𝑖=1 𝑞 𝐶 𝑖 𝑎 𝑖 + 𝑖=1 𝑞−1 𝑢 𝑚 𝑖≤𝑗≤𝑞 𝑤 𝑖𝑗 𝑑 𝑎 𝑖 , 𝑎 𝑗
+ 𝑗=𝑞+1 𝑀 𝐶 𝑗 𝑎 𝑗 + 𝑖=1 𝑞 𝑤 𝑖𝑗 𝑑 𝑎 𝑖 , 𝑎 𝑗
+ 𝑖=𝑞+1 𝑀−1 𝑗>𝑖 𝑤 𝑖𝑗 𝑑 𝑎 𝑖 , 𝑎 𝑗
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
TC 𝑎 𝑞 = 𝑖=1 𝑞 𝐶 𝑖 𝑎 𝑖 + 𝑖=1 𝑞−1 𝑢 𝑚 𝑖≤𝑗≤𝑞 𝑤 𝑖𝑗 𝑑 𝑎 𝑖 , 𝑎 𝑗
+ 𝑗=𝑞+1 𝑀 𝐶 𝑗 𝑎 𝑗 + 𝑖=1 𝑞 𝑤 𝑖𝑗 𝑑 𝑎 𝑖 , 𝑎 𝑗
+ 𝑖=𝑞+1 𝑀−1 𝑗>𝑖 𝑤 𝑖𝑗 𝑑 𝑎 𝑖 , 𝑎 𝑗
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
TC 𝑎 𝑞 = 𝑖=1 𝑞 𝐶 𝑖 𝑎 𝑖 + 𝑖=1 𝑞−1 𝑢 𝑚 𝑖≤𝑗≤𝑞 𝑤 𝑖𝑗 𝑑 𝑎 𝑖 , 𝑎 𝑗
+ 𝑗=𝑞+1 𝑀 𝐶 𝑗 𝑎 𝑗 + 𝑖=1 𝑞 𝑤 𝑖𝑗 𝑑 𝑎 𝑖 , 𝑎 𝑗
+ 𝑖=𝑞+1 𝑀−1 𝑗>𝑖 𝑤 𝑖𝑗 𝑑 𝑎 𝑖 , 𝑎 𝑗
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 w d34 w d35 w 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
G = g33 g g g36
g43 g g g46
g53 g g g56
g63 g g g66 +
w*d (assigned)
= Bound value of Node B 21

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

33. 3. Research Methods-Heuristic method
Why Bound value have problem?
WxD= [w41 w45 w46] x d34
d35
d36
G = g33 g g g36
g43 g g g46
g53 g g g56
g63 g g 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:
3. [fitness function]
Ex:f(213456) = 30

36. 3. Research Methods- Genetic Algorithm
4. [selection] selected for next generation
5. [crossover]
　If　S1=241356　S2=214563
then S1= S2=214356
6. [mutation]
Before mutation after mutation
7. Repeat 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 Bound value

38. 3. Research Methods- Heuristic method
(1．．．．．)
(2．．．．．)
(3．．．．．)
(4．．．．．)
(5．．．．．)
(6．．．．．)
(21．．．．)
(23．．．．)
(24．．．．)
(25．．．．)
(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
problem
method
Problem Size
(ex:3*3 6*6)
Symmetrical
Flow Size
Distance
…..
Spreadsheets
Branch and bound
A heuristic method

42. 5. Conclusion
This study proposes a heuristic method based on branch and bound algorithm for 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