SOLVING SET PARTITIONING PROBLEMS USING LAGRANGIAN RELAXATION

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1 SOLVING SET PARTITIONING PROBLEMS USING LAGRANGIAN RELAXATION Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Tilburg, op gezag van de rector magnificus, prof. dr. F.A. van der Duyn Schouten, in het openbaar te verdedigen ten overstaan van een door het college van promoties aangewezen commissie in de aula van de Universiteit op woensdag 15 maart 2006 om door Maria Gertruda Cornelia van Krieken geboren op 3 januari 1979 te 's-hertogenbosch.

2 Contents 1. Introduction The set partitioning problem Complexity Applications Literature Heuristics Optimal solution algorithms Test instances Goal and motivation of the research Outline of the thesis 8 2. Preprocessing Introduction Preprocessing rules for the set partitioning problem Equal k-columns Equal rows k-rowsets and contained rows Clique rule Cut rule Row combination technique Technique Implementation Row combination technique as preprocessing rule Individual computational results Equal k-columns k-rowsets and contained rows Clique and equal rows Cut and equal rows Row combination technique Links between the different techniques Relationship between contained rows and clique techniques Relationship between the contained rows and row combinations techniques Relationship between the cut and clique rules Relationship between k-rowset and clique 26

3 2.5.5 Relationship between the cut and 3-rowset rules Combined computational results Concluding remarks Lower bounds Theoretical background Linear programming relaxation Lagrangian relaxation Induced subproblems Lower bounds for induced subproblems Subgradient search Subgradient search methods Classic subgradient search Volume algorithm Static convergent series Dynamic convergent series Bundle dynamic convergent series Computational results Classic subgradient search Volume algorithm Static convergent series Dynamic convergent series Bundle dynamic convergent series Comparison Dual Heuristics Simple improvement heuristic OPT dual heuristic Computational results Lower bounds in LaRSS Reduced cost fixing Methods used in LaRSS for determination of lower bounds Lower bound results of LaRSS Concluding remarks Upper bounds Literature on heuristics Primal heuristic Implementation Row ordering Computational results Fixed row ordering Variable row ordering Concluding remarks Branch and bound Introduction 63

4 5.1.1 Literature Fathoming Introduction to this chapter Classical variable-based branching Constraint-based branching Static constraint-based branching Dynamic constraint-based branching Computational results Variable-based branching Static constraint-based branching Dynamic constraint-based branching Enhancing the branch and bound procedure Two difficult instances Dual update heuristic during branch and bound Dual 3OPT heuristic during branch and bound Lagrangian relaxation during branch and bound Concluding remarks Miscellaneous research results Cuts Clique inequalities Two heuristics for determining clique cuts Computational results Concluding remarks Decomposition approach Basic concept Problem formulation Computational experiments An alternative decomposition approach LaRSS Construction of LaRSS Computational results Technical aspects Efficiency Data management Concluding remarks Case study: collection of liquids coming from end-of-life vehicles Introduction Literature Model Planning methodology Route generation The route selection problem Validation and verification 113

5 8.4 Case results Scenario data Base scenario with partial and full collection of can-orders Sensitivity analysis on the length of collection period Set partitioning results Base scenario Double truck capacity Review period of three weeks General statistics Concluding remarks Business perspective Mathematical perspective Case study: vehicle routing in the closed-loop container network of ARN Introduction Background Goal Problem formulation: the 2-container collection problem Literature Methodology Route generation Route selection Structure of the analysis Simulation Data and scenarios Case results Current logistic network Network with uniform lifting mechanism for containers Set partitioning results Concluding remarks Business perspective Mathematical perspective Extensions Set packing constraints Preprocessing Lagrangian relaxation and dual heuristics Primal heuristic Branch and bound Other adjustments Computational results Set partitioning with side-constraints Preprocessing Lagrangian relaxation and dual heuristics Primal heuristic Branch and bound 154

6 Other adjustments Computational results Concluding remarks Epilogue Summary Preprocessing Lower bounds Upper bounds Branch and bound LaRSS Case studies Extensions Conclusion Conclusion on the performance of LaRSS Contributions Recommendations 166 A. Preprocessing implementations 169 A.1 Equal columns 169 A.2 Equal 2-columns 170 A.3 Equal k-columns 171 A.4 Equal rows 171 A.5 Contained rows 172 A.6 Equal 3-rowsets 173 A.7 Clique 174 A.8 Cut 175 References 177 Samenvatting 183 About the author 187