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The Travelling Salesman Problem a brief survey

The Travelling Salesman Problem a brief survey. Martin Grötschel Summary of Chapter 2 of the class Polyhedral Combinatorics (ADM III) May 18, 2010. Contents. Introduction The TSP and some of its history The TSP and some of its variants Some applications Modeling issues Heuristics

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The Travelling Salesman Problem a brief survey

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  1. The Travelling Salesman Problema brief survey Martin Grötschel Summary of Chapter 2 of the classPolyhedral Combinatorics (ADM III)May 18, 2010

  2. Contents • Introduction • The TSP and some of its history • The TSP and some of its variants • Some applications • Modeling issues • Heuristics • How combinatorial optimizers do it Martin Grötschel

  3. Contents • Introduction • The TSP and some of its history • The TSP and some of its variants • Some applications • Modeling issues • Heuristics • How combinatorial optimizers do it Martin Grötschel

  4. Combinatorial optimization Given a finite set E and a subset I of the power set of E (the set of feasible solutions). Given, moreover, a value (cost, length,…) c(e) for all elements e of E. Find, among all sets in I, a set I such that its total value c(I) (= sum of the values of all elements in I) is as small (or as large) as possible. The parameters of a combinatorial optimization problem are: (E, I, c). An important issue: How is I given? Martin Grötschel

  5. Special „simple“ combinatorial optimization problems Finding a • minimum spanning tree in a graph • shortest path in a directed graph • maximum matching in a graph • a minimum capacity cut separating two given nodes of a graph or digraph • cost-minimal flow through a network with capacities and costs on all edges • … These problems are solvable in polynomial time. Martin Grötschel

  6. Special „hard“ combinatorial optimization problems • travelling salesman problem (the prototype problem) • location und routing • set-packing, partitioning, -covering • max-cut • linear ordering • scheduling (with a few exceptions) • node and edge colouring • … These problems are NP-hard(in the sense of complexity theory). Martin Grötschel

  7. The travelling salesman problem Given n „cities“ and „distances“ between them. Find a tour (roundtrip) through all cities visiting every city exactly once such that the sum of all distances travelled is as small as possible. (TSP) The TSP is called symmetric (STSP) if, for every pair of cities i and j, the distance from i to j is the same as the one from j to i, otherwise the problem is called aysmmetric (ATSP). Martin Grötschel

  8. http://www.tsp.gatech.edu/

  9. THE TSPbook suggested reading for everyone interested in the TSP Martin Grötschel

  10. The travelling salesman problem Two mathematical formulations of the TSP • Does that help solve the TSP? Martin Grötschel

  11. Contents • Introduction • The TSP and some of its history • The TSP and some of its variants • Some applications • Modeling issues • Heuristics • How combinatorial optimizers do it Martin Grötschel

  12. Usually quoted as the forerunner of the TSP Usually quoted as the origin of the TSP Martin Grötschel

  13. about 100yearsearlier

  14. By a proper choice andscheduling of the tour onecan gain so much time that we have to makesome suggestions The most important aspect is to cover as many locations as possiblewithout visiting alocation twice Martin Grötschel

  15. Ulysses roundtrip (an even older TSP ?) The paper „The Optimized Odyssey“ by Martin Grötschel and Manfred Padberg is downloadable from http://www.zib.de/groetschel/pubnew/paper/groetschelpadberg2001a.pdf Martin Grötschel

  16. Ulysses The distance table Martin Grötschel

  17. Ulysses roundtrip optimal „Ulysses tour“ Martin Grötschel

  18. Malen nach ZahlenTSP in art ? • When was this invented? Martin Grötschel

  19. Survey Books Literature: more than 1000 entries in Zentralblatt/Math Zbl 0562.00014Lawler, E.L.(ed.); Lenstra, J.K.(ed.); Rinnooy Kan, A.H.G.(ed.); Shmoys, D.B.(ed.)The traveling salesman problem. A guided tour of combinatorial optimization. Wiley-Interscience Series in Discrete Mathematics. A Wiley-Interscience publication. Chichester etc.: John Wiley \& Sons. X, 465 p. (1985). MSC 2000: *00Bxx90-06 Zbl 0996.00026Gutin, Gregory (ed.); Punnen, Abraham P.(ed.)The traveling salesman problem and its variations. Combinatorial Optimization. 12. Dordrecht: Kluwer Academic Publishers. xviii, 830 p. (2002). MSC 2000: *00B1590-0690Cxx Martin Grötschel

  20. Contents • Introduction • The TSP and some of its history • The TSP and some of its variants • Some applications • Modeling issues • Heuristics • How combinatorial optimizers do it Martin Grötschel

  21. The Travelling Salesman Problem and Some of its Variants • The symmetric TSP • The asymmetric TSP • The TSP with precedences or time windows • The online TSP • The symmetric and asymmetric m-TSP • The price collecting TSP • The Chinese postman problem (undirected, directed, mixed) • Bus, truck, vehicle routing • Edge/arc & node routing with capacities • Combinations of these and more Martin Grötschel

  22. http://www.densis.fee.unicamp.br/~moscato/TSPBIB_home.html Martin Grötschel

  23. Contents • Introduction • The TSP and some of its history • The TSP and some of its variants • Some applications • Modeling issues • Heuristics • How combinatorial optimizers do it Martin Grötschel

  24. Production of ICs and PCBs Printed Circuit Board (PCB) Integrated Circuit (IC) Problems: Logical Design, Physical Design Correctness, Simulation, Placement of Components, Routing, Drilling,... Martin Grötschel

  25. Correct modelling of a printed circuit board drilling problem length of a move of the drilling head: Euclidean norm, Max norm, Manhatten norm? 2103 holes to be drilled Martin Grötschel

  26. Drilling 2103 holes into a PCB Significant Improvements via TSP (due to Padberg & Rinaldi) industry solution optimal solution Martin Grötschel

  27. Siemens-ProblemPCB da4 Martin Grötschel, Michael Jünger, Gerhard Reinelt,Optimal Control of Plotting and Drilling Machines: A Case Study, Zeitschrift für Operations Research, 35:1 (1991) 61-84 http://www.zib.de/groetschel/pubnew/paper/groetscheljuengerreinelt1991.pdf before after

  28. Siemens-Problem PCB da1 Grötschel, Jünger, Reinelt after before

  29. Martin Grötschel

  30. Leiterplatten-BohrmaschinePrinted Circuit Board Drilling Machine Martin Grötschel

  31. Foto einer Flachbaugruppe (Leiterplatte) Martin Grötschel

  32. Foto einer Flachbaugruppe (Leiterplatte) - Rückseite Martin Grötschel

  33. 442 holes to be drilled Martin Grötschel

  34. Typical PCB drilling problems at Siemens Table 4 Martin Grötschel

  35. Fast heuristics Table 5 Martin Grötschel

  36. Optimizing the stacker cranes of a Siemens-Nixdorf warehouse Martin Grötschel

  37. Herlitz at Falkensee (Berlin) Martin Grötschel

  38. Example: Control of the stacker cranes in a Herlitz warehouse Martin Grötschel

  39. Logistics of collectingelectronics garbage Andrea Grötschel Diplomarbeit (2004) Martin Grötschel

  40. Location plus tour planning (m-TSP) Martin Grötschel

  41. The Dispatching Problem at ADAC:an online m-TSP Dispatching Center (Pannenzentrale) Data Transm. „Gelber Engel“ Dispatcher Martin Grötschel

  42. Online-TSP (in a metric space) where Instance: 0 0 Goal: Find fastest tour serving all requests (starting and ending in 0) Algorithm ALG is c-competitive if for all request sequences

  43. Implementation competitions Martin Grötschel

  44. Contents • Introduction • The TSP and some of its history • The TSP and some of its variants • Some applications • Modeling issues • Heuristics • How combinatorial optimizers do it Martin Grötschel

  45. LP Cutting Plane Approach Even MODELLING is not easy! What is the „right“ LP relaxation? N. Ascheuer, M. Fischetti, M. Grötschel, „Solving the Asymmetric Travelling Salesman Problem with time windows by branch-and-cut“, Mathematical Programming A (2001), see http://www.zib.de/groetschel/pubnew/paper/ascheuerfischettigroetschel2001.pdf Martin Grötschel

  46. IP formulation of the asymmetric TSP Martin Grötschel

  47. Time Windows • This is a typical situation in delivery problems. • Customers must be served during a certain period of time, usually a time interval is given. • access to pedestrian areas • opening hours of a customer • delivery to assembly lines • just in time processes Martin Grötschel

  48. Model 1 Martin Grötschel

  49. Model 2 Martin Grötschel

  50. Model 3 Martin Grötschel

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