1. Presented by Yi-Tzu, Chen
A Lagrangian based approach for the asymmetric generalized traveling salesman problem Operations Research, 39(4): , 1991.
Presented by Yi-Tzu, Chen
2019/2/24
OPLab, Dept. of IM, NTU

2. Author C. E. Noon J. C. Bean. The University of Tennessee
Routing , scheduling of transportation systems and GIS for decision support
J. C. Bean.
The University of Michigan
GA, integer programming ,infinite horizon optimization, capacity expansion, production and scheduling
the Management Science the application of Geographic Information Systems
Noon 最佳化與應用的問題
Department of Industrial and Operations Engineering bean
2019/2/24
OPLab, Dept. of IM, NTU

3. Outline Introduction Optimal approach Computational Results Conclusion
Problem definition
Applications
Optimal approach
Bounds for the GTSP
Arc/Node Elimination
Enumeration procedure
Computational Results
Conclusion
Reflection
2019/2/24
OPLab, Dept. of IM, NTU

4. Introduction Problem definition i j Asymmetric and symmetric costij=20
costji=20
costji=10
Asymmetric
Symmetric
2019/2/24
OPLab, Dept. of IM, NTU

5. Introduction GTSP (Generalization Traveling salesman problem)
Directed graph
Grouped mutually exclusive
Exhaustive node sets
All cities were visited once
returns to the starting city
Min total cost
2019/2/24
OPLab, Dept. of IM, NTU

6. Introduction GTSP Definition , , , , , 2019/2/24
, , ,
, ,
2019/2/24
OPLab, Dept. of IM, NTU

7. Introduction
GTSP
Find a minimum cost m-arc cycle which include exactly one node from each node set
要找到一條min cost的路徑 通過每個group 而且是以同一個node 進出group
郵筒設置位置讓郵差方便
2019/2/24
OPLab, Dept. of IM, NTU

8. Introduction Applications Postal routing
Warehouse order picking with multiple stock locations
Airport selection and routing for courier planes
Reference:
[1]Laporte, G., H. Mercure and Y. Nobert, Generalized traveling salesman problem through n sets of nodes: the asymmetric case. Discrete appl. Math.18, ,1987
[2]Rousseau, J., Customization versus a general pourpose code for routing and scheduling problems: A point of view. In vehicle routing: method and studies, b. golden and A. Assad (eds.)Elsevier, Amsterdam, , 1988
[3] G. Laporte, A. Asef-Vaziri and C. Srikandarajah, “Some Applications of the Generalized Traveling Salesman Problem,” J. Opnl. Res. Soc. 47, , 1996.
倉庫 飛機停哪 郵筒設置的問題
2019/2/24
OPLab, Dept. of IM, NTU

9. Outline Introduction Optimal approach Computational Results Conclusion
Problem definition
Applications
Optimal approach
Bounds for the GTSP
Arc/Node Elimination
Enumeration procedure
Computational Results
Conclusion
Reflection
2019/2/24
OPLab, Dept. of IM, NTU

10. Optimal approach Problem P (1) Exactly one arc enter/leave a set
(2)Ensure the trip that is uninterrupted and continuous
(3)Prevent sub tours
目標是就是min the trip 的cost
在(1)的部分，這兩條限制式確保每個set只有一條arc進 跟只有一條arc 出來
(2)這條確立所有的node 是uninterrupted and continuous ，對每個點而言M，有近就一定有出，這樣整個trip就不會斷掉 ()
(4)Integer programming
2019/2/24
OPLab, Dept. of IM, NTU

11. Optimal approach Bounds for the GTSP Problem PRλ
λi ,λj is a vector of multiplier
2019/2/24
OPLab, Dept. of IM, NTU

12. Optimal approach Bounds for the GTSP i j k l .
Lemma 1 : There exists no optimal solution to PRλ in which xij=1, i∈ Si , j∈ Sj ,and , k∈ Si , j∈ Sj
進出node的都檢掉，如果有cost kl 比cost ij 小的話 那kl就是lower bound，用這方式找到最低的 i
costij=20 j k l
costkl=10
2019/2/24
OPLab, Dept. of IM, NTU

13. Optimal approach Bounds for the GTSP
Definition: Diagraph …is an aggregation of the original diagraph Let consist of m nodes where each node corresponds to a node set of
Lemma 2 : If x is a feasible solution to PRλover , then there exists a feasible TSP tour y over with
因lemma已經將一些set I 到set arc大的都取消掉了，所以現在GTSP轉成TSP
2019/2/24
OPLab, Dept. of IM, NTU

14. Optimal approach Bounds for the GTSP P->PRλ= GTSP->TSP 2019/2/24
OPLab, Dept. of IM, NTU

15. Optimal approach Bounds for the GTSP Problem APRλ
Subtour elimination (3) constrains of PRλ is dropped
1974年geoffrion有根沒有都不會影響到 以前適用direct method 現在是用LR
把sub tour relax調
2019/2/24
OPLab, Dept. of IM, NTU

16. Optimal approach Bounds for the GTSP Lower bound APRλ ≦ PRλ ≦ P
Upper bound
Nearest neighbor heuristic
m×m Assignment problem
m city TSP
Full GTSP
(N nodes, m groups )
1974年geoffrion有根沒有都不會影響到 以前適用direct method 現在是用LR
APR lower bound
2019/2/24
OPLab, Dept. of IM, NTU

17. Optimal approach Arc/Node Elimination 1.Arc Elimination
3.Nearest neighbor heuristic
Only iterated at most three times
可以跟arc一樣多次，但是實際上操作只有三次
2019/2/24
OPLab, Dept. of IM, NTU

18. Optimal approach Arc/Node Elimination j i Arc Elimination
Reduce cost :the increasing cost of an arc that is used in a solution
, : the cost that enter and leave Si
SET！！！！ j i
2019/2/24
OPLab, Dept. of IM, NTU

19. Optimal approach Arc/Node Elimination j i Arc Elimination Theorem :
, then the variable xij and its corresponding arc cannot be included in a feasible solution which is better than the current upper bound, and can be eliminated from the problem. j i
2019/2/24
OPLab, Dept. of IM, NTU

20. Optimal approach Arc/Node Elimination Node Elimination
Nodes that have no arcs
Theorem : For a given node I and any set Sk such that i ∉ Sk , if
then node I cannot be include in a feasible solution which is better than the current upper bound, and can be eliminated from the problem.
2019/2/24
OPLab, Dept. of IM, NTU

21. Optimal approach Enumeration procedure
By choosing a node set and branching
If a node included in the solution then fixing the other nodes of its set to be excluded.
represent a subproblem with first k node sets, and node i1 of Si…
2019/2/24
OPLab, Dept. of IM, NTU

22. Optimal approach Enumeration procedure Solve upper bound ( )and Stop!
lower bound
( )
Stop!
Subproblem
K+1=m
Formulate
PRλ for
subproblem
Arc/Node
Elimination
2019/2/24
OPLab, Dept. of IM, NTU

23. Outline Introduction Optimal approach Computational Results Conclusion
Problem definition
Applications
Optimal approach
Bounds for the GTSP
Arc/Node Elimination
Enumeration procedure
Computational Results
Conclusion
Reflection
2019/2/24
OPLab, Dept. of IM, NTU

24. Computational Results
2019/2/24
OPLab, Dept. of IM, NTU

25. Outline Introduction Optimal approach Computational Results Conclusion
Problem definition
Applications
Optimal approach
Bounds for the GTSP
Arc/Node Elimination
Enumeration procedure
Computational Results
Conclusion
Reflection
2019/2/24
OPLab, Dept. of IM, NTU

26. Conclusion Arc/node elimination reduce the problems
Bounding , elimination and enumeration enhance over GTSP
2019/2/24
OPLab, Dept. of IM, NTU

27. Outline Introduction Optimal approach Computational Results Conclusion
Problem definition
Applications
Optimal approach
Bounds for the GTSP
Arc/Node Elimination
Enumeration procedure
Computational Results
Conclusion
Reflection
2019/2/24
OPLab, Dept. of IM, NTU

28. Reflection GTSP->TSP Node elimination is effective?
GTSP turns fully into TSP
Split node!
2019/2/24
OPLab, Dept. of IM, NTU

29. Reflection
GTSP
TSP
2019/2/24
OPLab, Dept. of IM, NTU

30. Thanks for your listening
Q&A
Thanks for your listening
2019/2/24
OPLab, Dept. of IM, NTU