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A New Spatial Approach for Efficient Transformation of Equality - Generalized TSP to TSP

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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.

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A New Spatial Approach for Efficient Transformation of Equality - Generalized TSP to TSP

  1. 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. 2. TSP or Travelling Salesman Problem?
  3. 3. TSP or Travelling Salesman Problem
  4. 4. TSP or Travelling Salesman Problem
  5. 5. TSP or Travelling Salesman Problem
  6. 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. 7. Equality Generalized Travelling Salesman Problem?
  8. 8. Equality Generalized Travelling Salesman Problem
  9. 9. Equality Generalized Travelling Salesman Problem Cost Matrix nth nth
  10. 10. Equality Generalized Travelling Salesman Problem Solution?
  11. 11. Equality Generalized Travelling Salesman Problem Solution? Brute Force Heuristic Methods
  12. 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. 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. 14. What is Cost?
  15. 15. What is Cost? In Vehicle Navigation Instances 1. Distance 2. Time 3. Fuel Consumption 4. Scenic Beauty 5. Any Hybrid
  16. 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. 17. Equality - Generalized TSP with Static/Dynamic Costs ? ? ? ? ? ? ? ? ? ? ? ?
  18. 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. 19. Equality - Generalized TSP for Vehicle Navigation Dynamic Costs – Dynamic Data – Cost Matrix on the fly Static Costs – Static Data – Pre-processing possible
  20. 20. Equality - Generalized TSP for Vehicle Navigation 2 Problems in Pre-Processing (a) What is Distance? Dijkstra Distance / Displacement.
  21. 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. 22. Equality - Generalized TSP for Vehicle Navigation Distance = Dijkstra Distance or Displacement? 35,000 measurements from 210 cities Dijkstra Distance Displacement
  23. 23. Equality - Generalized TSP for Vehicle Navigation ? ? ? ? ? ? ? ? ? ? ? ? Displacement
  24. 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. 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. 26. Equality - Generalized TSP for Vehicle Navigation How to reduce Cost Matrix Size?
  27. 27. Equality - Generalized TSP for Vehicle Navigation
  28. 28. Equality - Generalized TSP for Vehicle Navigation
  29. 29. Equality - Generalized TSP for Vehicle Navigation
  30. 30. Equality - Generalized TSP for Vehicle Navigation
  31. 31. Equality - Generalized TSP for Vehicle Navigation 1 2 3 4 56 7 8 9 10 11 12 X Y
  32. 32. Equality - Generalized TSP for Vehicle Navigation 1 2 3 4 56 7 8 9 10 11 12 X Y
  33. 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. 34. Equality - Generalized TSP for Vehicle Navigation Each Cluster = [Product Cost1, Product Cost2, Product Cost3, . . . . ., Product Costp] Sorted List
  35. 35. Equality - Generalized TSP for Vehicle Navigation 5% Each Cluster = [Product Cost1, Product Cost2, Product Cost3, . . . . ., Product Costp] Sorted List
  36. 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. 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. 38. Equality - Generalized TSP for Vehicle Navigation 2 Problems in Pre-Processing (a) What is Distance? (b) Matrix size is very big
  39. 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. 40. Equality - Generalized TSP for Vehicle Navigation Average % Standard Deviation of Cluster’s Cost Error % Presented Approach GLKH Approach
  41. 41. Equality - Generalized TSP for Vehicle Navigation # of Clusters in Sample Instance Time taken (s) Presented Approach GLKH Approach
  42. 42. Equality - Generalized TSP for Vehicle Navigation # of Nodes in Sample Instance Matrix Size Presented Approach GLKH Approach
  43. 43. Equality - Generalized TSP for Vehicle Navigation # of Nodes in Sample Instance Matrix Size with value ≠ 0 or 9999 Presented Approach GLKH Approach
  44. 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. 45. Equality - Generalized TSP for Vehicle Navigation Gain in Time & Space – Polynomial Order Lose in Error – Logarithmic Order
  46. 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!

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