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Combinatorial Optimization in Action Martin Grötschel MathAcrossCampus Colloquium University of Washington Seattle, January 22, 2009 Martin Grötschel Institut für Mathematik, Technische Universität Berlin (TUB) DFG-Forschungszentrum “Mathematik für Schlüsseltechnologien” (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) [email protected] http://www.zib.de/groetschel 2 Contents 1. Where do I come from? 2. CO in Action: Solving problems from industry 3. What optimization problems can we solve well today? 4. Where are modelling challenges? 5. Where are mathematical challenges? 6. Public Transport 7. Logistics 8. Telecommunication 9. Summary Martin Grötschel 3 Contents 1. Where do I come from? 2. CO in Action: Solving problems from industry 3. What optimization problems can we solve well today? 4. Where are modelling challenges? 5. Where are mathematical challenges? 6. Public Transport 7. Logistics 8. Telecommunication 9. Summary Martin Grötschel 4 TU Berlin: last week Martin Grötschel 5 Konrad-Zuse-Zentrum Martin Grötschel 6 Some MATHEON statistics Financial Funding (in million Euros, 1€ ~ 1.3$) DFG 5.7 TU, HU, FU, ZIB, WIAS 3.1 State of Berlin 0.1 Technology Foundation Berlin 0.1 Total budget: 9.0 (plus industry support ,which varies from year to year) Grant period: 4+4+4 years, subject to successful peer review Funded Persons (2+2+2) ~ 40 professors & 7 junior research groups (3+2+2) ~160 researchers 70 researchers 20 research students 6 full professors DFG Research Center Matheon Persons involved 7 Festival of Mathematics 2006 Martin Grötschel Approximately 1000 participants, Audimax, TU Berlin, on November 16, 2006 8 The MATHEON Buddy Bear (June 11, 2005) Workshop BAG SPNV 9 existing DFG Research Centers rcom Ocean margins, Bremen (2001) CFN Functional nanostructures, Karlsruhe (2001) Rudolf-Virchow-Center Experimental biomedicine, Würzburg (2001) MATHEON Mathematics for key technologies, Berlin (2002) CMPB Molecular physiology of the brain, Göttingen (2002) CRTD Regenerative therapies, Dresden (2006) Martin Grötschel 10 MATHEON research I Optimization and discrete mathematics W. Römisch (HU) M. Skutella (TU) G. M. Ziegler (TU) II Numerical analysis and scientific computing R. Kornhuber (FU) R. Schneider (TU) H. Yserentant (TU)) III Applied and stochastic analysis C. Lasser (FU), A. Mielke (HU, WIAS) including: scientists in charge Martin Grötschel Mathematical fields 11 MATHEON research A Application areas Life sciences A. Bockmayr (FU), P. Deuflhard (FU, ZIB), Ch. Schütte (FU) Martin Grötschel B Logistics, traffic and telecommunication networks M. Grötschel (TU, ZIB), R. Möhring (TU), M. Skutella (TU) C Production C. Carstensen (HU), D. Hömberg (TU, WIAS), F. Tröltzsch (TU) D Circuit Simulation and Opto-Electronic Devices V. Mehrmann (TU), A. Mielke (HU,WIAS), F. Schmidt (ZIB) E Finance D. Becherer (HU), P. Imkeller (HU) F Visualization K. Polthier (ZIB), J. Sullivan (TU), G. M. Ziegler (TU) G Education, Outreach, Administration J. Kramer (HU) including: scientists in charge 12 The MATHEON Vision: Application Area B The role of networks Networks are omnipresent Rapidly growing in size and importance Their design and operation poses new challenges What constitutes a good network? Study common mathematical properties of network applications Develop theory, algorithms, and software for an advanced level of network analysis Address network planning problems as a whole Martin Grötschel 13 Contents 1. Where do I come from? 2. CO in Action: Solving problems from industry 3. What optimization problems can we solve well today? 4. Where are modelling challenges? 5. Where are mathematical challenges? 6. Public Transport 7. Logistics 8. Telecommunication 9. Summary Martin Grötschel The Problem Solving Cycle in Modern Applied Mathematics The Real Problem Hard- Software ware Data GUI Implementation in Practice Numerical Solution Simulation Modelling Quick Check: Heuristics Simulation The Application Driven Approach Practitioner Specialist Education Mathematical Model Optimization Algorithmic Implementation Computer Science Design of Good Solution Algorithms Mathematical Theory Pure Mathematics 15 Beginning with the End, a School Activity: Combinatorial Optimization in Education Education Film Martin Grötschel 16 No good application without adequate theory! Discrete Mathematics 1. Graph Theory 2. Matroids and Indepentent Sets 3. Min-Max Results 4. Combinatorial Theory 5. Polyhedral Combinatorics Optimization 1. Linear Programming 2. Duality 3. Nonlinear Programming 4. Nonlinear Nondifferential Programming 5. 0/1-Programming 6. Integer Programming 7. Mixed-Integer Programming 8. Stochastic Programming E.ON-Project on gas transport and gas pipeline capacity planning 9. Stochastic Nonlinear MixedInteger Programming since January 2009 1 million Euros/year 8. Online Optimization 9. Complexity Theory 10. Understanding Heuristics Martin Grötschel 17 Contents 1. Where do I come from? 2. CO in Action: Solving problems from industry 3. What optimization problems can we solve well today? 4. Where are modelling challenges? 5. Where are mathematical challenges? 6. Public Transport 7. Logistics 8. Telecommunication 9. Summary Martin Grötschel 18 typical optimization problems min cT x h j ( x) 0, j 1, 2,..., m min cT x Ax a Bx b x0 x R n (and x S ) ( x R n n) x j Z, j J max f ( x) or min f ( x) gi ( x) 0, i 1, 2,..., k ( x k n n) „general“ (nonlinear) program NLP linear program LP program = optimization problem Martin Grötschel Ax a Bx b x0 ( x j 0,1) (linear) mixedinteger program IP, MIP 20 The Simplex Method for linear programming Dantzig, 1947: primal Simplex Method Lemke, 1954; Beale, 1954: dual Simplex Method Dantzig, 1953: revised Simplex Method …. Underlying Idea: Find a vertex of the set of feasible LP solutions (polyhedron) and move to a better neighbouring vertex, if possible. Martin Grötschel 21 The Simplex Method: an example min/max + x1 + 3x2 (1) (2) (3) (4) (5) - x2 - x1 - x2 - x1 + x2 + x1 + x1 + 2x2 <= 0 <=-1 <= 3 <= 3 <= 9 (4) (1) Martin Grötschel 22 The Simplex Method: an example min/max + x1 + 3x2 (1) (2) (3) (4) (5) - x2 - x1 - x2 - x1 + x2 + x1 + x1 + 2x2 <= 0 <=-1 <= 3 <= 3 <= 9 (4) (1) Martin Grötschel 23 computationally important idea of the Simplex Method Let a (m,n)-Matrix A with full row rank m, an m-vector b and an n-vector c with m<n be given. For every vertex y of the polyhedron of feasible solutions of the LP, max cT x Ax b N A= B x0 there is a non-singular (m,m)-submatrix B (called basis) of A representing the vertex y (basic solution) as follows yB B 1b, yN 0 Martin Grötschel Many computational consequences: Update-formulas, reduced cost calculations, number of non-zeros of a vertex,… 24 Hirsch Conjecture If P is a polytope of dimension n with m facets then every vertex of P can be reached from any other vertex of P on a path of length at most m-n. In the example before: m=5, n=2 and m-n=3, conjecture is true. At present, not even a polynomial bound on the path length is known. Best upper bound: Kalai, Kleitman (1992): The diameter of the graph of an n-dimensional polyhedron with m facets is at most m(log n+1). Lower bound: Holt, Klee (1997): at least m-n (m, n large enough). Martin Grötschel 25 The Ellipsoid Method Shor, 1970 - 1979 Yudin & Nemirovskii, 1976 Khachiyan, 1979 M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms and Combinatorial Optimization Algorithms and Combinatorics 2, Springer, 1988 Martin Grötschel 26 The Ellipsoid Method: an example Martin Grötschel Initialization Stopping criterion Feasibility check Cutting plane choice Update The Ellipsoid Method 29 Interior-Point Methods: an example Karmarkar (1984) and many others afterwards Often also called Barrier Methods central path (4) (1) min Martin Grötschel interior Point George Dantzig and Bob Bixby, 2000 Linear Programming ILOG slides obtained from Bob Bixby on November 11, 2007 © Copyright 2006 - ILOG, Inc. - All Rights Reserved 30 LP Progress: An Example A Production Planning Model 401,640 cons. 1,584,000 vars. 9,498,000 nonzeros Solution time line (2.0 GHz P4): 1988 (CPLEX 1.0): 1997 (CPLEX 5.0): 2003 (CPLEX 9.0): 29.8 days 1.5 hours 59.1 seconds Speedup 1x 480x 43500x Solving IN 1988: 82 years (machines 1000x slower) © Copyright 2006 - ILOG, Inc. - All Rights Reserved 31 LP Progress: 1988-2004 (Operations Research, Jan 2002, pp. 3—15, updated in 2004) Algorithms (machine independent): Primal versus best of Primal/Dual/Barrier Machines (workstations PCs): NET: Algorithm × Machine 3300x 1600x 5 300 000x (2 months/5300000 ~= 1 second) Bad news: Little change in LP since 2004 © Copyright 2006 - ILOG, Inc. - All Rights Reserved 32 33 Why do I talk about LP? Algorithms to solve linear programs are the most important ingredient of the techniques for solving combinatorial and integer programming problems. Martin Grötschel 34 The Branch&Bound Technique: An Example min cT x Ax a Bx b x0 min cT x Ax a Bx b x0 x 0,1 n 0/1program Martin Grötschel x 0,1 n x 1 LPrelaxation Solve the LP-relaxation and get optimal solution y. (lower bound) If y integral, DONE! Otherwise pick fractional component y(i). Create two new subproblems by adding y(i)=1 and y(i)=0, resp. LP solution …. y(i)=0 y(k)=1 integral solution = upper bound y(i)=1 = lower bound y(j)=1 35 Branching (in general) Current solution is infeasible Martin Grötschel 36 Branching (in general) Rounding a fractional component up and down Decomposition into subproblems removes infeasible solution Martin Grötschel 37 cutting plane technique for integer and mixed-integer programming Feasible integer solutions Objective function Convex hull LP-based relaxation Cutting planes Martin Grötschel Mixed Integer Speedups 1991-2008 George Dantzig and Ralph Gomory the fathers of Linear Programming and Integer Programming 40 Primal and Dual Heuristics Primal Heuristic: Finds a (hopefully) good feasible solution. Dual Heuristic: Finds a bound on the optimum solution value (e.g., by finding a feasible solution of the LP-dual of an LP-relaxation of a combinatorial optimization problem). Minimization: dual heuristic value ≤ optimum value ≤ primal heuristic value (In maximization the inequalities are the other way around.) quality guarantee Martin Grötschel in practice and theory 41 Primal and Dual Heuristics Primal and Dual Heuristics give rise to worst-case guarantee: Minimization: optimum value ≤ ≤ dual heuristic value ≤ ≤ primal heuristic value (1+e) optimum value primal heuristic value (1+e) dual heuristic value (In maximization the inequalities are the other way around.) quality guarantee Martin Grötschel in practice and theory 42 Martin Grötschel Linear & Mixed-Integer Programming: ZIB Software-Suite SCIP: Constrained Mixed Integer Programming (Verwendung z.B. in Bereichen Verkehr, Telekommunikation, TU Berlin, TU Darmstadt, TU Chemnitz, Universität Bayreuth, ..., auch bei CO@Work) SoPlex- und SCIP-Weiterentwicklung, SCIP-SoPlex-Integration (Kooperation mit TU Darmstadt; Finanzierung durch Siemens) ILOG/ZIB-Kooperation zu Forschung in mathematischer Programmierung Modellierungssprache ZIMPL (mit Anbindung an SCIP) perPlex: LP-Prüfer mit exakter Arithmetik Schnittebenenverfahren, MIP-Primalheuristiken Branch-and-Bound (z. B. Reliability-Branching, Conflict Analysis) Testbibliotheken MIPLIB, MADLIB, SteinLib, … 43 Martin Grötschel A ZIB/MATHEON contribution to CPLEX 44 General computational results linear programming: excellent nonlinear programming: problem specific integer programming: problem specific mixed-integer programming: very problem specific nonlinear mixed-integer programming: very very problem specific Stochastic nonlinear mixed-integer programming: very very very problem specific Martin Grötschel 45 Contents 1. Where do I come from? 2. CO in Action: Solving problems from industry 3. What optimization problems can we solve well today? 4. Where are modelling challenges? 5. Where are mathematical challenges? 6. Public Transport 7. Logistics 8. Telecommunication 9. Summary Martin Grötschel 46 The “classical” Transportation Problem in Mathematical Programming min ( i , j )S T cij xij subject to x ai i S S = sources, origins, supply x b j j T T = sinks, destinations, demand jT iS ij ij 0 xij ( capij ) Martin Grötschel This problem rarely occurs in real life in its pure form. It does appear as a subproblem of some much more complex real problems. It can be solved very quickly. 47 Just one example: Planning Public Transportation Phase: Planning Scheduling Dispatching Horizon: Long Term Medium term (very) Short term Timetable Period Day of Operation online planning Objective: Service Level Cost Reduction Get it done Steps: Network Design Line Planning Timetabling Vehicle Scheduling Duty Scheduling Duty Rostering Crew Assignment Delay Management Failure Management politically difficult done well Martin Grötschel industry not ready 48 Contents 1. Where do I come from? 2. CO in Action: Solving problems from industry 3. What optimization problems can we solve well today? 4. Where are modelling challenges? 5. Where are mathematical challenges? 6. Public Transport 7. Logistics 8. Telecommunication 9. Summary Martin Grötschel 49 New challenges with each new application very, very large-scale linear programs – see integrated scheduling The curse of symmetry – see GSM channel assignment Breaking NP-completeness in practice Understand new structures – nonlinear constraints in MIP Employ new insights from other areas – semi-algebraic a side step follows geometry. Multi-scales – railroad rolling stock circulation Multi-objective optimization – almost everywhere Online and real time optimization – ADAC, harbour Martin Grötschel 50 Semi-algebraic Geometry Real-algebraic Geometry fi ( x), g j ( x), hk ( x) are polynomials in d real variables dd S : {x R : f1 ( x) 0,..., fl ( x) 0} dd basic closed S : {x R : g1 ( x) 0,..., g m ( x) 0} basic open dd S : {x R : h1 ( x) 0,..., hn ( x) 0} S : S Martin Grötschel S S is a semi-algebraic set 51 Theorem of Bröcker(1991) & Scheiderer(1989) basic closed case Every basic closed semi-algebraic set of the form dd S {x R : f1 ( x) 0,..., fl ( x) 0}, where fi R[ x1 ,..., xd ],1 i l , are polynomials, can be represented by at most d (d 1) / 2 polynomials, i.e., there exist polynomials p1 ,..., pd ( d 1) / 2 R[ x1 ,..., xd ] such that S {x Rd : p1 ( x) 0,..., pd ( d 1) / 2 ( x) 0}. Martin Grötschel 52 Our main theorem Theorem Let P R n be a n-dimensional polytope given by an inequality representation. Then 2n polynomials pi R[ x1 ,..., xn ] can be constructed such that P P (p1 ,..., p2n ). Can this insight be used algorithmically? Hartwig Bosse, Martin Grötschel, Martin Henk: Polynomial inequalities representing polyhedra Mathematical Programming 103 (2005)35-44 Further results by Averkov&Henk and Bröcker Martin Grötschel 53 Contents 1. Where do I come from? 2. CO in Action: Solving problems from industry 3. What optimization problems can we solve well today? 4. Where are modelling challenges? 5. Where are mathematical challenges? 6. Public Transport 7. Logistics 8. Telecommunication 9. Summary Martin Grötschel 54 How do graphs look? Martin Grötschel Petersen graph 55 How do graphs look? Martin Grötschel Petersen graph 56 Vehicle Scheduling in Public Transit large cases: 100 million edges = # of variables Martin Grötschel 57 Duty Scheduling in Public Transit large cases: 1,000 million edges = # of variables Martin Grötschel 58 Airline Crew Scheduling large cases: 10 million edges = # of variables plus additional constraints Martin Grötschel 59 Planning in Public Transport Cost Recovery Fares Construction Costs Network Topology Velocities Lines Service Level Frequencies Connections Timetable Sensitivity Rotations Relief Points Duties Duty Mix Rostering Fairness Crew Assignment Disruptions Operations Control IS-OPT Martin Grötschel APD VS-OPT2 DS-OPT VS-OPT BS-OPT AN-OPT B15 B1 B3 B1 multidepartmental Departments multidepotwise Depots multiple line groups Line Groups multiple lines Lines multiple rotations Rotations 60 Solution “Technology” linear programming integer and mixed-integer programming nondifferentiable optimization lots of heuristics Martin Grötschel 61 BVG (Berlin): bus circulation Martin Grötschel 62 Vehicle Scheduling: The "Camel Curve" 68 Martin Grötschel Savings in Berlin public transport 64 Integer Programming Model 700 000 deadheads VSP 1 000 000 duties DSP coupling constraints Martin Grötschel 28 000 cons 6 000 cons 150 000 cons 65 Solving Very Large LPs (IVU41 838,500 x 3,570, 10.5 NNEs per column) 450 400 350 300 250 0 20 40 60 80 100 [s] Martin Grötschel Coordinate Ascent Subgradient Volume Bundle+AS Dual Simplex Barrier 66 Bundle Method (Kiwiel [1990], Helmberg [2000]) max f ( ) max min c x vehicleschedule T ) x 1Dx min d T y duty schedule y 2Cy ) piecewise linear functions quadratic subproblem (dualization reduces dim.) fˆ f1 primal & dual convergence approximation f 2 3 1 decomposable functions f ( ) c T x T (b Ax ) Martin Grötschel fˆk ( ) : min f ( ) Jk uk ˆ k 1 argmax fk ( ) ˆk 2 2 67 Integrated Scheduling 1983-2007 Article trips v d Problem sequential planning Ball et al. [1983] 1 1 000 -- 133 Scott [1985] 1 456 54 -- VSP + duty cost estimate 17 300 -- -- VSP + additional constraints Falkner & Ryan [1992] 1 182 -- 41 DSP + additional constraints Patrikalakis et al. [1992] -- 111 20 45 DSP + min cost flow 28 257 44 65 ISP without driver releases Freling [1997] 1 296 38 90 ISP Friberg & Haase [1997] 1 30 -- -- Freling et al. [2000] 1 476 9 23 ISP Huisman [2004] -- 653 67 117 ISP Weider [2007] 7 3 698 209 260 ISP + caps + resource cons Tosini & Vercellis [1988] Gaffi & Nonato [1997] Martin Grötschel depots DSP + SPP to optimality 68 Modelling challenge: two examples Mathematical layout design of transportation infrastructure public transportation systems of cities/regions making best use of railway tracks Martin Grötschel 69 Origin-Destination Matrix of Potsdam Martin Grötschel 70 Determining ticket prices in public transport Ansatz „Controlling“ demand via prices and travel times Price system = Individual price + ??? Maximize profit? Maximize user utility? Electronic Ticketing Status Data? Mathematical models ? Giant amount of economic literature on topics of questionable value for practice (versions of local elasticities) Martin Grötschel 71 Planning prices optimal price income/profit demand price Martin Grötschel Maximize income price 73 Fare – Demand – Revenue Demand/pax Fare/€ Demand Martin Grötschel d = d(x) Revenue/€ Fare/€ Revenue r(x) = d(x) · p(x) 74 Demand Functions Logit vs. constant elasticity e(x) := x·d'(x)/d(x) (d/d)/(x/x) Curtin [1968] : e – 0.3 Cobb-Douglas-functions d(x) = c·xe with constant elasticity Martin Grötschel 75 Example: Single and Monthly Ticket (FPP) max p iC' stD s.t. Martin Grötschel i st ( x) d sti ( x) xP xs = 1.57 € / 1.79 € xm = 43.72 € / 48.21 € r(x) = 2,129,971 € (+3%) d(x) = 58,601.4 (-12%) Modal split = 28% (-4%) 76 Comparing Fare Systems Status quo fare system revenue demand cost single/monthly 2 072 106 66 503.0 3 597 604 single/monthly 1 659 052 30 200.6 1 500 000 standard/reduced 1 428 374 43 942.6 1 500 000 single/monthly 2 053 676 66 723.6 3 553 676 standard/reduced 1 675 575 73 474.5 3 175 575 Max-P* Max-D* * = Subsidy 1 500 000 € Martin Grötschel 77 Service Planning in Public Transport Fares Martin Grötschel Demand Lines Transfers Timetable 78 Railway network as a market place The railway network manager is obliged by EU and German law to offer as much network capacity as possible to all train operation companies (TOCs) non-discriminatingly → The network is a market place, but not a simple one Help needed to impove the market design Martin Grötschel 79 Our sample network Martin Grötschel 80 Results Test Network 45 Tracks 32 Stations 6 Traintypes 10 Trainsets 122 Nodes 659 Arcs 3-12 Hours 96 Station Capacities 612 Headway Times Martin Grötschel 81 The project Trassenbörse: Railway Slot Auctioning The project aims at developing new ideas to make better (or even best) use of railway tracks. A basic assumption, always favoured by economists, is that "markets" lead to an optimal allocation of goods. But what are the goods to be allocated in the "railway market"? And if we can define such goods precisely, how can one introduce trade mechanisms that lead to fair competition? In other words, is there a way to deregulate the current railway system that results in a "better" utilization of the railway infrastructure? Martin Grötschel 82 Difficulties to be considered What is a slot precisely? How many details can/should be taken into account? What about track profiles? What about engine characteristics? Routing through stations? Weichengenaue Planung (switch scheduling)? Buffers and slacks? Signals? Auctioning process Martin Grötschel 83 The project Trassenbörse: Railway Slot Auctioning The collection of question raised calls for a multidisciplinary approach. The project is carried out by a group of economists, mathematicians, and railway engineers from Berlin and Hannover, each group bringing its particular expertise. Martin Grötschel 84 Rail Track Auction TOCs decide on bids for slots BEGIN Bid is unchanged Bid is increased by a minimum increment yes All bids Unchanged? no OPTRA model is solved with maximum earnings no yes Wish to increase bid? yes no END Martin Grötschel Bid assigned? 85 Modelling challenges Mathematical models that help making best use of transportation infrastructure Designing auctions for technically difficult infrastructure use Martin Grötschel 86 Contents 1. Where do I come from? 2. CO in Action: Solving problems from industry 3. What optimization problems can we solve well today? 4. Where are modelling challenges? 5. Where are mathematical challenges? 6. Public Transport 7. Logistics 8. Telecommunication 9. Summary Martin Grötschel 87 Optimization of Botany Bay a Container Terminal in Sydney, Australia Martin Grötschel 88 Martin Grötschel Port Botany – existing Terminal layout 89 MATHEON Project B14: Combinatorial Aspects of Logistics Task: Practice: nothing like the logistics problem (specific aspects in each application) Wanted: identify and tackle core models/approaches, e.g. reoptimization for online problems Applications: Martin Grötschel online/offline control of logistics systems vehicle dispatching, elevator control, automated transportation systems, ... 90 B14: Recent Applications Scheduling of laser welding robots in car body manufacturing (Volkswagen): During welding each robot is fed by a laser source Goal: minimize number of required laser sources Control of destination-call elevator systems (Kollmorgen Steuerungstechnik): Passenger specifies destination already when calling an elevator Goal: small average/maximal waiting and journey times Martin Grötschel 91 Contents 1. Where do I come from? 2. CO in Action: Solving problems from industry 3. What optimization problems can we solve well today? 4. Where are modelling challenges? 5. Where are mathematical challenges? 6. Public Transport 7. Logistics 8. Telecommunication 9. Summary Martin Grötschel 92 The beginning: Clyde Monma (Bell Labs/Bell Communications Research) Cornell University, 1987 Survivable telecommunications networks What was the problem? Martin Grötschel 93 Martin Grötschel The BellCore study 94 Real data Martin Grötschel 95 Problemreductions Martin Grötschel 96 Computational results with real data Martin Grötschel 97 LATA DL: optimal solutions Martin Grötschel 98 Problem Nobody at Bell was interested (except for the scientists). We were too much ahead of time! But then! Martin Grötschel 99 But then: USA 1987-1988 (collected by Clyde Monma) Martin Grötschel 100 Survivability-Models: today still a hot topic mathematical models and software for: Diversification „route node-disjoint“ H B 120 D H 60 D F 30 F 30 M Reservation „reroute all demands“ M H (or p% of all affected demands) B H 60 D (or p% of all demands) Path restoration „reroute affected demands“ B D 60 F 120 F M H M B H 60 D F 60 D B 60 F M plus: simultaneous capacity planning and routing Martin Grötschel B 60 M 101 Mathematical Model Te min k et x et e E t 1 x et {0,1} e E , t 1, xet 1 xet e E, t 1, Te ye cet xet ,Te , Te topology decisison capacity decisions normal operation routing component failure routing e E t 0 ye 0 uvD Puv :eP duv Martin Grötschel 0 Puv 0 fuv (P) e E 0 fuv (P) s fuv (P) 0 uv D s s S, uv Ds ,P uv 102 Finding feasible solutions? Heuristics Manipulation of Local search – Routings – Topology – Capacities Simulated Annealing Genetic algorithms ... Problem Sizes Martin Grötschel Nodes Edges Demands Routing-Paths 15 46 78 > 150 x 10e6 36 107 79 > 500 x 10e9 36 123 123 > 2 x 10e12 103 What can one save? Real scenario • • • PhD Thesis: 163 nodes http://www.zib.de/wessaely 227 edges [email protected] 561 demands 34% potential savings! == > hundred million dollars Martin Grötschel 104 Getting more general: What is the Telecom Problem? Design excellent technical devices and a robust network that survives all kinds of failures and organize the traffic such that high quality telecommunication between very many individual units at many locations is feasible at low cost! Martin Grötschel Speech Data Video Etc. 105 What is the Telecom Problem? Design excellent technical devices and a robust network that survives all kinds of failures and organize the traffic such that high quality telecommunication between very many individual units at many locations is feasible at low cost! Martin Grötschel This problem is too general to be solved in one step. Approach in Practice: Decompose whenever possible. Look at a hierarchy of problems. Address the individual problems one by one. Recompose to find a good global solution. 106 Connecting Mobiles: What´s up? BSC MSC MSC BSC BSC MSC MSC BSC BSC MSC BSC BTS Martin Grötschel BSC 107 Frequency or Channel Assignment The story to be told now is based on GSM technology (GSM = Global System for Mobile Communications) There are other mobile communication technologies such as UMTS (UMTS = Universal Mobile Telecommunications System), a system that is based on CDMA technology (CDMA = Code Division Multiple Access) where the „story“ is different. Martin Grötschel 108 Antennas & Interference co- & adjacent channel interference cell antenna x x x x site x x x x cell backbone network Martin Grötschel 109 Interference Level of interference depends on distance between transmitters geographical position power of the signals direction in which signals are transmitted weather conditions assigned frequencies co-channel interference adjacent-channel interference Martin Grötschel 110 Separation Frequencies assigned to the same location (site) have to be separated Site Blocked channels Parts of the spectrum forbidden at some locations: government regulations, agreements with operators in neighboring regions, requirements military forces, etc. Martin Grötschel 111 FAP: Frequency Assignment Problem Find an assignment of frequencies to transmitters that satisfies all separation constraints all blocked channels requirements and either avoids interference at all or minimizes the (total/maximum) interference level Martin Grötschel 112 Minimum Interference Frequency Assignment Problem (FAP) FAP is an Integer Linear Program: min vwE s.t. co co cvw zvw f Fv vwE ad co x vf 1 xvf xwg 1 co xvf xwf 1 zvw ad xvf xwg 1 zvw co ad xvf , zvw , zvw 0,1 Martin Grötschel ad ad cvw zvw v V vw E d , f g d (vw) vw E co , f Fv Fw vw E ad , f g 1 that is very difficult to solve. 113 [% mi ni m ] av um era de ge gre de e ma gr ee xim um de dia gr me ee cli ter qu en um be r de ns |V | Ins tan ity ce A Glance at some Instances k 267 56,8 2 151,0 B-0-E-20 1876 13,7 40 257,7 f 2786 4,5 3 135,0 h 4240 5,9 11 249,0 238 779 453 561 E-Plus Project Martin Grötschel 3 69 5 81 12 69 10 130 114 Region Berlin - Dresden 2877 carriers 50 channels Interference reduction: 83.6% Martin Grötschel 115 UMTS Movie (UMTS auction: the source of MATHEON funds) Movie Martin Grötschel 116 G-WiN Data G-WiN = Gigabit-Wissenschafts-Netz of the DFN-Verein Internet access of all German universities and research institutions Locations to be connected: 750 Data volume in summer 2000: 220 Terabytes/month Expected data volume in 2004: 10.000 Terabytes/month Clustering (to design a hierarchical network): 10 nodes in Level 1a 261 nodes eligible for 20 nodes in Level 1b Level 1 All other nodes in Level 2 Martin Grötschel 117 G-WiN Location/Clustering Problem min wip xip pZ iV x ip 1 pZ k K p wkp z kp Each location i must be connected to a Level 1 node p di xip i k z p 1 k k c z p p Capacity at p must be large enough k Only one configuration at each Location 1 node k k z p const p All variables are 0/1. Martin Grötschel # of Level 1a nodes 118 Solution: Hierarchy & Backbone Martin Grötschel 119 G-WiN Location Problem: Solution Statistics The DFN problem leads to ~100.000 0/1-variables. Typical computational experience: Optimal solution via CPLEX in a few seconds! A very related problem at Telekom Austria has ~300.000 0/1-variables plus some continuous variables and capacity constraints. Computational experience (before problem specific fine tuning): 10% gap after 6 h of CPLEX computation, 60% gap after „simplification“ (dropping certain capacities). Martin Grötschel 120 X-WIN G-WIN served the ~750 scientific institutions from 2000 to 2006. G-WIN was reconfigured about every two months to meet changes in demand. Three modifications were allowed at each update at most. With new transport, hub, and switching technologies new design possibilities arise. We have designed the new German science network, called X-WIN which started operating at the end of 2006 (terabit backbone, 10 gigabit/second connections,…) Martin Grötschel 121 Data and a glimpse at the model initial model: 1 billion variables after reduction ~100.000 variables ~100.000 constraints solved by ZIMPL/CPLEX in a few minutes. 81 scenarios have been considered and solved – after lots of trials – for each choice of reasonable number of core nodes. Martin Grötschel 122 Number of Nodes in the Core Network Martin Grötschel 123 Contents 1. Where do I come from? 2. CO in Action: Solving problems from industry 3. What optimization problems can we solve well today? 4. Where are modelling challenges? 5. Where are mathematical challenges? 6. Public Transport 7. Logistics 8. Telecommunication 9. Summary Martin Grötschel 124 Mathematical challenges (examples coming from public transport) solving integrated models multicommodity flow (vehicle circulation, 100 million variables) set partitioning (driver scheduling, 1 billion variables) combined solution at present hopeles for large instances multi-objective integer programs minimize number of vehicles minimize operation costs minimize environmental impact (CO2, energy consumption) minimize number of drivers minimize driver costs all at the same time nonlinear integer programming Martin Grötschel pricing 125 Mathematical challenges stochastic integer programs finding „realistic“ distributions (stochastic models) solving stochastic integer programs real time/online optimization solving rescheduling problems of real instances in real time multi-scale integer programming example: integrating various time scales and scales of technical detail train scheduling (day-week) track assignment in stations (minutes) rolling stock maintainance (weeks-months) Martin Grötschel 126 Mathematical challenges mathematics of regulation/deregulation railway track auctioning slot management in airports energy A major issue in the new E.ON gas transport project Nobel Prize in Economics 2007 auctioning mechanism design Martin Grötschel 127 Summary Large improvement potentials in telecommunication with respect to cost survivability capacity congestion interference …. New technologies bring new questions, e.g., NGN-networks Martin Grötschel 128 Summary Large improvement potentials in logistics, (public) transport with respect to cost pollution, environmental impact service quality management support/quality quality control …. Can public transport break even? Where are the bottlenecks? Martin Grötschel Book Presentation on November 11, 2008 Thanks for Combinatorial Optimization your attention in Action Martin Grötschel MathAcrossCampus Colloquium University of Washington Seattle, January 22, 2009 Martin Grötschel Institut für Mathematik, Technische Universität Berlin (TUB) DFG-Forschungszentrum “Mathematik für Schlüsseltechnologien” (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) [email protected] http://www.zib.de/groetschel