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The opportunity cost of time window violations

  • Matteo Salani
  • Maria Battarra
Research Paper

Abstract

This paper studies a variant of the vehicle routing problem with soft time windows (VRPSTW), inspired by real-world distribution problems. In applications, violations of the prescribed delivery time are commonly accepted. Customers’ inconvenience due to early or late arrival is typically modelled as a penalty cost included in the VRPSTW objective function, added to the routing costs. However, weighting routing costs against customer inconvenience is not straightforward for practitioners. In our problem definition, practitioners evaluate solutions by comparison with the hard time windows solution. The desired routing cost saving is set by the practitioners as a percentage of the nominal solution’s routing costs. The objective function minimizes the time window violations, or the customer inconvenience, with respect to the nominal solution. This allows practitioners to quantify the opportunity cost (i.e. the customer inconvenience), when a target routing cost saving is imposed. To solve the problem, we apply two exact algorithms: the first is based on a standard branch-and-cut-and-price (BCP), the second is a BCP nested in a bisection algorithm. Computational results demonstrate that the second algorithm outperforms the standard implementation. Solutions obtained with the opportunity cost interpretation of soft time windows are then compared with solutions obtained using both hard time windows and the standard interpretation of soft time windows.

Keywords

Vehicle routing Soft time windows Branch-and-cut-and-price Oppourtunity cost 

References

  1. Balakrishnan N (1993) Simple heuristics for the vehicle routeing problem with soft time windows. J Oper Res Soc 44:279–287CrossRefGoogle Scholar
  2. Baldacci R, Mingozzi A, Roberti R (2012) Recent exact algorithms for solving the vehicle routing problem under capacity and time window constraints. Eur J Oper Res 218:1–6CrossRefGoogle Scholar
  3. Bettinelli A, Ceselli A, Righini G (2014) A branch-and-price algorithm for the multi-depot heterogeneous-fleet pickup and delivery problem with soft time windows. Math Progr Comput 6(2):171–197CrossRefGoogle Scholar
  4. Bhusiri N, Qureshi A, Taniguchi E (2014) The trade-off between fixed vehicle costs and time-dependent arrival penalties in a routing problem. Transp Res Part E Logist Transp Rev 62:1–22CrossRefGoogle Scholar
  5. Bräysy O, Gendreau M (2005a) Vehicle routing problem with time windows, part I: route construction and local search algorithms. Transp Sci 39:104–118CrossRefGoogle Scholar
  6. Bräysy O, Gendreau M (2005b) Vehicle routing problem with time windows, part II: metaheuristics. Transp Sci 39:119–139CrossRefGoogle Scholar
  7. Calvete H, GalT C, Oliveros M, Sánchez-Valverde B (2004) Vehicle routing problems with soft time windows: an optimization based approach. Monografías del Seminario Matemático García de Galdeano 31:295–304Google Scholar
  8. Caramia M, Dell’ Olmo P (eds) (2008) Multi-objective management in freight logistics. Springer London, LondonGoogle Scholar
  9. Chang WC, Russell R (2004) A met aheuristic for the vehicle-routeing problem with soft time windows. J Oper Res Soc 55:1298–1310CrossRefGoogle Scholar
  10. Cordeau JF, Desaulniers G, Desrosiers J, Solomon M, Soumis F (2002) VRP with time windows. In: Toth P, Vigo D (eds) The vehicle routing problem, chap 7. SIAM, Philadelphia, pp 157–193CrossRefGoogle Scholar
  11. Dantzig GB, Ramser JH (1959) The truck dispatching problem. Manag Sci 6:80–91CrossRefGoogle Scholar
  12. Desaulniers G, Desrosiers G, Dumas Y, Solomon M, Soumis F (1997) Daily aircraft routing and scheduling. Manag Sci 43(6):841–855CrossRefGoogle Scholar
  13. Desaulniers G, Desrosiers J, Solomon M (eds) (2005) Column generation. GERAD 25th anniversary series. Springer, BerlinGoogle Scholar
  14. Desaulniers G, Desrosiers J, Spoorendonk S (2010) The vehicle routing problem with time windows: state-of-the-art exact solution methods. In: Cochran J, Cox L, Keskinocak P, Kharoufeh J, Smith J (eds) Wiley encyclopedia of operations research and management science. Wiley, New YorkGoogle Scholar
  15. Desaulniers G, Madsen O, Ropke S (2014) VRP with time windows. In: Toth P, Vigo D (eds) Vehicle routing: problems, methods, and applications. SIAM, PhiladelphiaGoogle Scholar
  16. Dumas Y, Soumis F, Desrosiers J (1990) Optimizing the schedule for a fixed vehicle path with convex inconvenience costs. Transp Sci 24:145–152CrossRefGoogle Scholar
  17. Fagerholt K (2001) Ship scheduling with soft time windows: an optimisation based approach. Eur J Oper Res 131:559–571CrossRefGoogle Scholar
  18. Ferland J, Fortin L (1989) Vehicles scheduling with sliding time windows. Eur J Oper Res 38:213–226CrossRefGoogle Scholar
  19. Figliozzi M (2010) An iterat ive route construction and improvement algorithm for the vehicle routing problem with soft time windows. Transp Res Part C Emerg Technol 18:668–679CrossRefGoogle Scholar
  20. Fu Z, Eglese R, Li L (2008) A unified tabu search algorithm for vehicle routing problems with soft time windows. J Oper Res Soc 59:663–673CrossRefGoogle Scholar
  21. Gendrau M, Tarantilis C (2010) Solving large-scale vehicle routing problems with time windows: the state-of-the-art. Tech. Rep. 2010-04, CIRRELTGoogle Scholar
  22. Golden B, Raghavan S, Wasil EA (2008) The vehicle routing problem: latest advances and new challenges. Operations research/computer science interfaces series, vol 43. Springer, BerlinCrossRefGoogle Scholar
  23. Ibaraki T, Imahori S, Kubo M, Masuda T, Uno T, Yagiura M (2005) Effective local search algorithms for routing and scheduling problems with general time-window constraints. Transp Sci 39:206–232CrossRefGoogle Scholar
  24. Ibaraki T, Imahori S, Nonobe K, Sobue K, Uno T, Yagiura M (2008) An iterated local search algorithm for the vehicle routing problem with convex time penalty functions. Discrete Appl Math 156:2050–2069CrossRefGoogle Scholar
  25. Ioannou G, Kritikos M, Prastacos G (2003) A problem generator-solver heuristic for vehicle routing with soft time windows. Omega 31:41–53CrossRefGoogle Scholar
  26. Jabali O, Leus R, Van Woensel T, de Kok T (2015) Self-imposed time windows in vehicle routing problems. OR Spectr 37(2):331–352CrossRefGoogle Scholar
  27. Kallehauge B (2008) Formulations and exact algorithms for the vehicle routing problem with time windows. Comput Oper Res 35:2307–2330CrossRefGoogle Scholar
  28. Koskosidis Y, Powell W, Solomon M (1992) An optimization-based heuristic for vehicle routing and scheduling with soft time window constraints. Transp Sci 26:69–85CrossRefGoogle Scholar
  29. Liberatore F, Righini G, Salani M (2011) A column generation algorithm for the vehicle routing problem with soft time windows. 4OR Q J Oper Res 9:49–82CrossRefGoogle Scholar
  30. Min H (1991) A multiobjective vehicle routing problem with soft time windows: the case of a public library distribution system. Socio Econ Plan Sci 25:179–188CrossRefGoogle Scholar
  31. Pullen H, Webb M (1967) A compu ter application to a transport scheduling problem. Comput J 10:10–13CrossRefGoogle Scholar
  32. Qureshi A, Taniguchi E, Yamada T (2009) An exact solution approach for vehicle routing and scheduling problems with soft time windows. Transp Res Part E Logist Transp Rev 45:960–977CrossRefGoogle Scholar
  33. Righini A, Salani M (2008) New dynamic programming algorithms for the resource constrained shortest path problem. Networks 51:155–170CrossRefGoogle Scholar
  34. Righini G, Salani M (2006) Symmetry helps: bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints. Discrete Optim 3(3):255–273CrossRefGoogle Scholar
  35. Rousseau LM, Gendreau M, Feillet D (2007) Interior point stabilization for column generation. Oper Res Lett 35(5):660–668CrossRefGoogle Scholar
  36. Ruinelli L, Salani M, Gambardella LM (2012) Hybrid column generation-based approach for vrp with simultaneous distribution, collection, pickup-and-delivery and real-world side constraints. Proc ICORES 2012:247–255Google Scholar
  37. Salani M, Vacca I (2011) Branch and price for the vehicle routing problem with discrete split deliveries and time windows. Eur J Oper Res 213(3):470–477CrossRefGoogle Scholar
  38. Schrage L (1981) Formulation and structure of more complex/realistic routing and scheduling problems. Networks 11:229–232CrossRefGoogle Scholar
  39. Sexton T, Bodin L (1985a) Optimizing single vehicle many-to-many operations with desired delivery times: I. Scheduling. Transp Sci 19:378–410CrossRefGoogle Scholar
  40. Sexton T, Bodin L (1985b) Optimizing single vehicle many-to-many operations with desired delivery times: II. Routing. Transp Sci 19:411–435CrossRefGoogle Scholar
  41. Sexton T, Choi YM (1986) Pickup and delivery of partial loads with “soft” time windows. Am J Math Manag Sci 6:369–398Google Scholar
  42. Solomon M (1983) Vehicle routing and scheduling with time windows constraints: models and algorithms. PhD thesis, University of PennsylvaniaGoogle Scholar
  43. Taş D, Gendreau M, Dellaert N, van Woensel T, de Kok A (2014a) Vehicle routing with soft time windows and stochastic travel times: a column generation and branch-and-price solution approach. Eur J Oper Res 236(3):789–799CrossRefGoogle Scholar
  44. Taş D, Jabali O, Van Woensel T (2014b) A vehicle routing problem with flexible time windows. Comput Oper Res 52(Part A):39–54Google Scholar
  45. Taillard E, Badeau P, Gendreau M, Guertin F, Potvin JY (1997) A tabu search heuristic for the vehicle routing problem with soft time windows. Transp Sci 31:170–186CrossRefGoogle Scholar
  46. Toth P, Vigo D (2002) The vehicle routing problem. SIAM monographs on discrete mathematics and applications. Society for Industrial and Applied Mathematics, PhiladelphiaGoogle Scholar
  47. Vanderbeck F, Wolsey L (1996) An exact a lgorithm for ip column generation. Oper Res Lett 19:151–159CrossRefGoogle Scholar
  48. Vareias A, Repoussis P, Tarantilis C (2018) Assessing customer service reliability in route planning with self-imposed time windows and stochastic travel times. Transp Sci (To appear)Google Scholar
  49. Vidal T, Crainic T, Gendreau M, Prins C (2012) Heuristics for multi-attribute vehicle routing problems: a survey and synthesis. Tech. Rep. 2012-05, CIRRELTGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and EURO - The Association of European Operational Research Societies 2018

Authors and Affiliations

  1. 1.Dalle Molle Institute for Artificial Intelligence (IDSIA)USI/DTI-SUPSILuganoSwitzerland
  2. 2.School of ManagementUniversity of BathBathUK

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