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The Travelling Salesman Problem: A brief survey. Martin Grötschel Vorausschau auf die Vorlesung Das Travelling-Salesman-Problem (ADM III) im WS 2013/14 14. Oktober 2013. Contents. Introduction The TSP and some of its history The TSP and some of its variants Some applications
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The Travelling Salesman Problem:A brief survey Martin Grötschel Vorausschau auf die VorlesungDas Travelling-Salesman-Problem (ADM III) im WS 2013/1414. Oktober 2013
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
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
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
Special „simple“ combinatorial optimization problems Finding a • minimum spanning tree in a graph • shortest path in a directed graph • maximum matching in a graph • 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
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
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 asymmetric (ATSP). Martin Grötschel
THE TSPbook suggested reading for everyone interested in the TSP Martin Grötschel
Another recommendationBill Cook‘s new book Martin Grötschel
The travelling salesman problem Two mathematical formulations of the TSP • Does that help solve the TSP? Martin Grötschel
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
Usually quoted as the forerunner of the TSP Usually quoted as the origin of the TSP Martin Grötschel
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
A TSP contest 1962: 10.000 $ Prize Martin Grötschel
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
Ulysses The distance table Martin Grötschel
Ulysses roundtrip optimal „Ulysses tour“ Martin Grötschel
Malen nach ZahlenTSP in art ? • When was this invented? Martin Grötschel
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
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
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
http://www.densis.fee.unicamp.br/~moscato/TSPBIB_home.html Martin Grötschel
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
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
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
Drilling 2103 holes into a PCB Significant Improvements via TSP (due to Padberg & Rinaldi) industry solution optimal solution Martin Grötschel
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
Siemens-Problem PCB da1 Grötschel, Jünger, Reinelt after before
Leiterplatten-BohrmaschinePrinted Circuit Board Drilling Machine Martin Grötschel
Foto einer Flachbaugruppe (Leiterplatte) Martin Grötschel
Foto einer Flachbaugruppe (Leiterplatte) - Rückseite Martin Grötschel
442 holes to be drilled Martin Grötschel
Typical PCB drilling problems at Siemens Table 4 Martin Grötschel
Fast heuristics Table 5 Martin Grötschel
Optimizing the stacker cranes of a Siemens-Nixdorf warehouse Martin Grötschel
Herlitz at Falkensee (Berlin) Martin Grötschel
Example: Control of the stacker cranes in a Herlitz warehouse Martin Grötschel
Logistics of collectingelectronics garbage Andrea Grötschel Diplomarbeit (2004) Martin Grötschel
Location plus tour planning (m-TSP) Martin Grötschel
The Dispatching Problem at ADAC:an online m-TSP Dispatching Center (Pannenzentrale) Data Transm. „Gelber Engel“ Dispatcher Martin Grötschel
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
Implementation competitions Martin Grötschel
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
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
IP formulation of the asymmetric TSP Martin Grötschel
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
Model 1 Martin Grötschel