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The path programming problem and a partial path relaxation

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  • Dollevoet, T.A.B.
  • Pecin, D.
  • Spliet, R.

Abstract

We introduce the class of path programming problems, which can be used to model many known optimization problems. A path programming problem can be formulated as a binary programming problem, for which the pricing problem can be modeled as a shortest path problem with resource constraints when column generation is used to solve its linear programming relaxation. Many optimization problems found in the literature belong to this class. We provide a framework for obtaining a partial path relaxation of a path programming problem. Like traditional path relaxations, the partial path relaxation allows the computational complexity of the pricing problem to be reduced, at the expense of a weaker linear programming bound. We demonstrate the versatility of this framework by providing different examples of partial path relaxations for a crew scheduling problem and vehicle routing problem.

Suggested Citation

  • Dollevoet, T.A.B. & Pecin, D. & Spliet, R., 2020. "The path programming problem and a partial path relaxation," Econometric Institute Research Papers EI-2020-04, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    • Handle: RePEc:ems:eureir:127709
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    References listed on IDEAS

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    1. Roberto Baldacci & Aristide Mingozzi & Roberto Roberti, 2011. "New Route Relaxation and Pricing Strategies for the Vehicle Routing Problem," Operations Research, INFORMS, vol. 59(5), pages 1269-1283, October.
    2. Martin Desrochers & Jacques Desrosiers & Marius Solomon, 1992. "A New Optimization Algorithm for the Vehicle Routing Problem with Time Windows," Operations Research, INFORMS, vol. 40(2), pages 342-354, April.
    3. Remy Spliet & Adriana F. Gabor, 2015. "The Time Window Assignment Vehicle Routing Problem," Transportation Science, INFORMS, vol. 49(4), pages 721-731, November.
    4. Stefan Irnich & Daniel Villeneuve, 2006. "The Shortest-Path Problem with Resource Constraints and k -Cycle Elimination for k (ge) 3," INFORMS Journal on Computing, INFORMS, vol. 18(3), pages 391-406, August.
    5. Stefan Irnich & Guy Desaulniers, 2005. "Shortest Path Problems with Resource Constraints," Springer Books, in: Guy Desaulniers & Jacques Desrosiers & Marius M. Solomon (ed.), Column Generation, chapter 0, pages 33-65, Springer.
    6. Bach, Lukas & Gendreau, Michel & Wøhlk, Sanne, 2015. "Freight railway operator timetabling and engine scheduling," European Journal of Operational Research, Elsevier, vol. 241(2), pages 309-319.
    7. Spliet, Remy & Desaulniers, Guy, 2015. "The discrete time window assignment vehicle routing problem," European Journal of Operational Research, Elsevier, vol. 244(2), pages 379-391.
    8. Daniel Potthoff & Dennis Huisman & Guy Desaulniers, 2010. "Column Generation with Dynamic Duty Selection for Railway Crew Rescheduling," Transportation Science, INFORMS, vol. 44(4), pages 493-505, November.
    9. Dollevoet, T.A.B. & Munari, P. & Spliet, R., 2020. "A p-step formulation for the capacitated vehicle routing problem," Econometric Institute Research Papers EI2020-01, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    10. Guy Desaulniers & Fausto Errico & Stefan Irnich & Michael Schneider, 2016. "Exact Algorithms for Electric Vehicle-Routing Problems with Time Windows," Operations Research, INFORMS, vol. 64(6), pages 1388-1405, December.
    11. Moshe Dror, 1994. "Note on the Complexity of the Shortest Path Models for Column Generation in VRPTW," Operations Research, INFORMS, vol. 42(5), pages 977-978, October.
    12. Erwin Abbink & Matteo Fischetti & Leo Kroon & Gerrit Timmer & Michiel Vromans, 2005. "Reinventing Crew Scheduling at Netherlands Railways," Interfaces, INFORMS, vol. 35(5), pages 393-401, October.
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    Keywords

    Path programming; Partial path relaxation; Shortest path problem with resource constraints; Column Generation;
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