The Equality - Generalized Travelling Salesman Problem (E-GTSP) asks to find a Hamiltonian cycle visiting each group exactly once, where each group represents a type of visiting node. This can represent a range of combinatorial optimization problem of NP-hard type like planning, logistics, etc. Its solution requires transformation of E-GTSP to TSP before solving it using a given TSP solver. This paper presents 5 different search-algorithms for optimal transformation which considers spatial spread of nodes of each group. Algorithms are tested over 15 cities with different street-network’s fractal-dimension for 5 instances of group-counts each. It’s observed that the R-Search algorithm, which selects nodes from each group depending upon their radial separation with respect to the start-end point, is the optimal search criterion among all other algorithms with a mean length error of 8.8%. This study will help developers and researchers to answer complex routing problems from a spatial perspective.

1. A New Spatial Approach for Efficient Transformation of
Equality - Generalized TSP to TSP
Mohammed Zia
PhD in Geomatics Engineering
Istanbul Technical University, Turkey18th August 2017

2. TSP or Travelling Salesman Problem?

3. TSP or Travelling Salesman Problem

4. TSP or Travelling Salesman Problem

5. TSP or Travelling Salesman Problem

6. TSP or Travelling Salesman Problem
1
2
3
4
5
9999 24 45 56 98
24 9999 10 24 32
45 10 9999 65 54
56 24 65 9999 100
98 32 54 100 9999
1 2 3 4 5
1
2
3
4
5
Cost Matrix

7. Equality Generalized Travelling Salesman Problem?

8. Equality Generalized Travelling Salesman Problem

9. Equality Generalized Travelling Salesman Problem
Cost Matrix
nth
nth

10. Equality Generalized Travelling Salesman Problem
Solution?

11. Equality Generalized Travelling Salesman Problem
Solution?
Brute Force Heuristic Methods

12. Equality Generalized Travelling Salesman Problem
Heuristic Solution
Equality Generalized Travelling Salesman Problem Instance
Clustered Travelling Salesman Problem Instance
Travelling Salesman Problem Instance
SOLUTION
Reduce to
Reduce to
Solve TSP
Derive Solution

13. Equality Generalized Travelling Salesman Problem
Heuristic Solution
Charles E. Noon and James C. Bean, February 1993, An Efficient
Transformation Of The Generalized Traveling Salesman Problem, INFOR
Information Systems and Operational Research, Volume 30 (1), DOI:
10.1080/03155986.1993.11732212

14. What is Cost?

15. What is Cost?
In Vehicle Navigation Instances
1. Distance
2. Time
3. Fuel Consumption
4. Scenic Beauty
5. Any Hybrid

16. What is Cost?
In Vehicle Navigation Instances
1. Distance Static
2. Time Dynamic
3. Fuel Consumption Dynamic
4. Scenic Beauty Dynamic
5. Any Hybrid Static/Dynamic

17. Equality - Generalized TSP with Static/Dynamic Costs
?
?
? ? ?
?
?
? ? ?
?
?

18. Equality - Generalized TSP with Dynamic Costs
Equality Generalized Travelling Salesman Problem Instance
Clustered Travelling Salesman Problem Instance
Travelling Salesman Problem Instance
SOLUTION
Reduce to
Reduce to
Solve TSP
Derive Solution
Current Equality Generalized TSP Instance with Empty Cost Matrix
?

19. Equality - Generalized TSP for Vehicle Navigation
Dynamic Costs – Dynamic Data – Cost Matrix on the fly
Static Costs – Static Data – Pre-processing possible

20. Equality - Generalized TSP for Vehicle Navigation
2 Problems in Pre-Processing
(a) What is Distance? Dijkstra Distance / Displacement.

21. Equality - Generalized TSP for Vehicle Navigation
2 Problems in Pre-Processing
(a) What is Distance?
(b) Matrix size is very big Ex. Istanbul – 23,000 Amenities
Dijkstra Distance / Displacement.

22. Equality - Generalized TSP for Vehicle Navigation
Distance = Dijkstra Distance or Displacement?
35,000 measurements
from 210 cities
Dijkstra Distance
Displacement

23. Equality - Generalized TSP for Vehicle Navigation
?
?
? ? ?
?
?
? ? ?
?
?
Displacement

24. Equality - Generalized TSP for Vehicle Navigation
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA NA = 9999
1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
5
6
7
8
9
10
11
12

25. Equality - Generalized TSP for Vehicle Navigation
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA NA = 9999
1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
5
6
7
8
9
10
11
12
122 – (32 + 42 + 22 + 32)
or
𝑛2- 𝑚=1
𝑘
|𝑚|2

26. Equality - Generalized TSP for Vehicle Navigation
How to reduce Cost Matrix Size?

27. Equality - Generalized TSP for Vehicle Navigation

28. Equality - Generalized TSP for Vehicle Navigation

29. Equality - Generalized TSP for Vehicle Navigation

30. Equality - Generalized TSP for Vehicle Navigation

31. Equality - Generalized TSP for Vehicle Navigation
1
2
3
4
56
7
8
9
10
11
12
X
Y

32. Equality - Generalized TSP for Vehicle Navigation
1
2
3
4
56
7
8
9
10
11
12
X
Y

33. Equality - Generalized TSP for Vehicle Navigation
1
2
3
4
56
7
8
9
10
11
12
X
Y
Average Cost1
Average Cost3
Product Cost =
Avg. Cost1 x Avg. Cost2 x Avg. Cost3

34. Equality - Generalized TSP for Vehicle Navigation
Each Cluster = [Product Cost1, Product Cost2, Product Cost3, . . . . ., Product Costp]
Sorted List

35. Equality - Generalized TSP for Vehicle Navigation
5%
Each Cluster = [Product Cost1, Product Cost2, Product Cost3, . . . . ., Product Costp]
Sorted List

36. Equality - Generalized TSP for Vehicle Navigation
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA NA = 9999
1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
5
6
7
8
9
10
11
12
1 4 5 8 10 11
NA
NA
NA
NA
NA
NA
1
4
5
8
10
11

37. Equality - Generalized TSP for Vehicle Navigation
2 Problems in Pre-Processing
(a) What is Distance?
(b) Matrix size is very big Ex. Istanbul – 23,000 Amenities
Dijkstra Distance / Displacement.

38. Equality - Generalized TSP for Vehicle Navigation
2 Problems in Pre-Processing
(a) What is Distance?
(b) Matrix size is very big

39. Equality - Generalized TSP for Vehicle Navigation
GTSP Sample Instance Library
Download - http://www.cs.rhul.ac.uk/home/zvero/GTSPLIB/
Presented Approach Vs Generalized Lin–Kernighan–Helsgaun Approach
For GLKH State-of-the-Art Approach –
Keld Helsgaun, 2015, Solving the equality generalized traveling salesman
problem using the Lin–Kernighan–Helsgaun Algorithm, Mathematical
Programming Computation, Volume 7, Page 269–287, DOI 10.1007/s12532-015-
0080-8

40. Equality - Generalized TSP for Vehicle Navigation
Average % Standard Deviation of Cluster’s Cost
Error %
Presented
Approach
GLKH Approach

41. Equality - Generalized TSP for Vehicle Navigation
# of Clusters in Sample Instance
Time taken (s)
Presented
Approach
GLKH Approach

42. Equality - Generalized TSP for Vehicle Navigation
# of Nodes in Sample Instance
Matrix Size
Presented
Approach
GLKH Approach

43. Equality - Generalized TSP for Vehicle Navigation
# of Nodes in Sample Instance
Matrix Size
with value
≠ 0 or 9999 Presented
Approach
GLKH Approach

44. Average % Standard Deviation of Cluster’s Cost
Error %
# of Clusters in Sample Instance
Time
taken (s)
# of Nodes in Sample Instance
Matrix Size
# of Nodes in Sample Instance
Matrix Size
with value
≠ 0 or 9999
For 5%

45. Equality - Generalized TSP for Vehicle Navigation
Gain in Time & Space – Polynomial Order
Lose in Error – Logarithmic Order

46. World TSP tour
Length: 7,516,353,779km
Cities: 1,904,711
Source: http://www.math.uwaterloo.ca/tsp/world/pictures.html
Thank You!