INTEGER PROGRAMMING BASED SEARCH A Thesis Presented to The Academic Faculty by Michael R Hewitt In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Industrial and Systems Engineering Georgia Institute of Technology December 2009 INTEGER PROGRAMMING BASED SEARCH Approved by: Professor Martin Savelsbergh, Advisor Professor Ozlem Ergun School of Industrial and Systems School of Industrial and Systems Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Professor Alan Erera, Co-Advisor Professor Mark Ferguson School of Industrial and Systems College of Management Engineering Georgia Institute of Technology Georgia Institute of Technology Professor George Nemhauser,Co-Advisor Date Approved: August 2009 School of Industrial and Systems Engineering Georgia Institute of Technology ACKNOWLEDGEMENTS I would like to thank my advisors Alan Erera, George Nemhauser and Martin Savelsbergh for the unique perspective and expertise they each brought to this thesis and my graduate studies. I valued both greatly. I could not have asked for more from doing research with the three of them; it simply has been a lot of fun. I would especially like to thank Martin Savelsbergh for his involvement and patience with my maturation as a researcher and writer. I would like to thank the industry partners Yellow Roadway Corporation, Saia. Inc and Exxon Mobil Corporation, who have sponsored my research at one point or another. Each of them reminded and reassured me of the practical importance of the work I was doing. Finally, I’d like to thank my wife and daughter for letting me work so much but not all the time. Don’t worry. Tenure is only 6 years away! iii TABLE OF CONTENTS ACKNOWLEDGEMENTS ................................ iii LISTOFTABLES .................................... vi LISTOFFIGURES .................................... vii SUMMARY......................................... viii I INTRODUCTION.................................. 1 II IP-BASEDNEIGHBORHOODSEARCH. 11 2.1 Literature.................................... 13 2.2 Formulations .................................. 13 2.3 SolutionApproach............................... 15 2.3.1 NeighborhoodSearch . .. .. .. .. .. .. .. 16 2.3.2 LowerBounds ............................. 22 2.3.3 Initial Feasible Solution . 24 2.4 ComputationalResults . .. .. .. .. .. .. .. .. 25 2.4.1 InstanceGeneration . .. .. .. .. .. .. .. 26 2.4.2 Calibration............................... 27 2.4.3 UpperBound.............................. 27 2.4.4 LowerBound.............................. 36 III MODELINGIP-BASEDSEARCH. .. .. .. .. .. .. .. 38 3.1 Modeling the Search for Optimal Solutions . ..... 39 3.2 Solving Instance of MP,MP= ........................ 43 3.3 Multi-Commodity Fixed-Charge Network Flow . ..... 49 3.4 ComputationalResults . .. .. .. .. .. .. .. .. 50 3.4.1 PrimalSide............................... 52 3.4.2 DualSide................................ 56 3.4.3 ComparisonwithIPSearch. 57 IV IP-BASEDSEARCHINPRACTICE . 60 4.1 Background................................... 63 4.2 LiteratureReview ............................... 66 iv 4.3 EnhancedLoadPlanning ........................... 67 4.4 ModelingFreightRouting. 72 4.5 Load Plan Design Integer Program . 77 4.5.1 Variations on Traditional Load Plan Design . 78 4.6 In-tree Reoptimization Heuristic . ..... 81 4.7 ComputationalResults . .. .. .. .. .. .. .. .. 84 4.7.1 SolvingIn-treeIPs........................... 84 4.7.2 Traditional Load Plan Improvements . 87 4.7.3 Variations on the Traditional Load Plan . 89 V CONCLUSIONSANDFUTURERESEARCH . 93 REFERENCES....................................... 97 v LIST OF TABLES 1 Comparison with Tabu Search and Path Relinking . 29 2 Primal-sideComparisonwithCPLEX . 31 3 IPSearchGivenMoreTime........................... 32 4 Primal Side Comparison with CPLEX - Metaheuristic Instances - Single Path 34 5 Selection Method Contribution . 35 6 PrimalandDualComparisonwith defaultCPLEX . 54 7 Results by Number Commodities in an Instance . 55 8 Primal Comparison with CPLEX and Local Branching . 57 9 PrimalSideComparisonof4and6Processors . 58 10 PrimalandDualComparisonwithIPSearch . 59 11 OptimalityGapReductions . 85 12 Number IIPd Solvedtowithin.1%....................... 86 13 %Savingsinthirtyminutes . 86 14 LoadPlanSavingsForEachWeek . 88 15 LoadPlanSavingsForEachWeek . 88 16 CostComponentComparison . 88 17 Load Plan Savings For Load Plan Variants . 90 18 Load Plan Variants Cost Component Comparison . ..... 90 19 Allowing Different Paths on Different Days-Apr08 . ...... 90 20 Allowing Different Paths on Different Days - Mar09 . ..... 91 21 % Terminals That Load a Single Destination on Multiple Directs - Apr08 . 92 22 % Terminals That Load a Single Destination on Multiple Directs - Mar09 . 92 vi LIST OF FIGURES 1 Datasharingbetweenprocesses . 48 2 Primal-sidePerformanceOverTime . 56 3 #ShipmentsbyServiceStandard. 61 4 ExampleNetwork................................. 69 5 Freight routing decisions yield the following trailer movements when empty costsarenotconsidered.. 70 6 Freight routing decisions made in conjunction with empty repositioning de- cisions yield higher loaded costs, but lower total costs. .......... 70 7 Load plan on Monday does not consolidate freight. ...... 71 8 Load plan on Tuesday-Friday consolidates freight through BB terminal b. 72 9 Holding Freight for Consolidation . 75 10 LN for a four terminal network, and load plan as a directed in-tree into d . 76 11 Time-space network depicting routing choices into d given a load plan. 77 12 Savingsovertime................................. 89 vii SUMMARY When integer programming (IP) models are used in operational situations there is a need to consider the tradeoff between the conflicting goals of solution quality and solution time, since for many problems solving realistic-size instances to a tight tolerance is still beyond the capability of state-of-the-art solvers. However, by appropriately defining small instances, good primal solutions frequently can be found quickly. We explore this approach in this thesis by studying the design of algorithms that produce solutions to an integer program by solving restrictions of the problem via integer programming technology. We refer to this type of algorithm as IP-based search. This approach is also taken, for example, within LP-based branch-and-bound algorithms using techniques such as Local Branching and Relaxation Induced Neighborhood Search (RINS). These techniques use information from the LP solution and incumbent solution to define a small IP, which is then optimized. These techniques can be applied to any integer program and are available in commercial solvers such as CPLEX. We develop new IP-based search approaches for specific problems that exploit problem structure and an approach that can be easily applied to general integer programs. Finally, we leverage some of the strengths of IP-based search to develop new and more accurate models of a network design problem faced by freight transportation carriers. In the first part of the thesis we present a heuristic for the classical Multi-Commodity Fixed Charge Network Flow (MCFCNF) model that exploits problem structure to produce high quality solutions quickly. The solution approach combines mathematical program- ming and heuristic search techniques. To obtain high-quality solutions it relies on neigh- borhood search with neighborhoods that involve solving carefully chosen integer programs derived from the arc-based formulation of MCFCNF. To obtain lower bounds, the linear programming relaxation of the path-based formulation is used and strengthened with cuts discovered during the neighborhood search. Computational experiments demonstrate that viii the proposed approach outperforms both best-known meta-heuristics and a state-of-the-art MIP solver. In the second part of the thesis we present an IP-based search algorithm for mixed integer programs that does not depend on problem structure. We formalize IP-based search as solving a restriction of the original problem and then develop an extended formulation to model the choice of restriction to solve. We propose a parallelized branch-and-price scheme for solving the extended formulation that is designed to produce high quality solutions quickly. We illustrate the application of the algorithm on the MCFCNF and computational experiments indicate it is competitive both with a state-of-the-art MIP solver and the structure-based heuristic presented in the previous part, even though it is a more general algorithm. Lastly, the thesis addresses the applicability of IP-based search to a real-world problem; namely the Service Network Design problem faced by Less-Than-Truckload (LTL) freight transportation carriers. In this part we present advances both in modeling and algorithm design. The developed models more accurately capture key operations of today’s carriers: decisions for loaded and empty trailer movements are considered simultaneously, and a time discretization is used that can appropriately model the timing of freight consolidation opportunities. Along with providing decision support for traditional service network plans used by LTL carriers, the models also enable the development of plans that allow more flexibility, such as allowing certain freight routes to vary by weekday. Given the additional detail within the proposed models, very large problem instances result when they are applied to large-scale LTL networks. Yet computational experiments using data from a large U.S.