, Volume 22, Issue 3, pp 1123–1147 | Cite as

A two-stage stochastic transportation problem with fixed handling costs and a priori selection of the distribution channels

  • Yolanda Hinojosa
  • Justo Puerto
  • Francisco Saldanha-da-GamaEmail author
Original Paper


In this paper, a transportation problem comprising stochastic demands, fixed handling costs at the origins, and fixed costs associated with the links is addressed. It is assumed that uncertainty is adequately captured via a finite set of scenarios. The problem is formulated as a two-stage stochastic program. The goal is to minimize the total cost associated with the selected links plus the expected transportation and fixed handling costs. A prototype problem is initially presented which is then progressively extended to accommodate capacities at the origins and multiple commodities. The results of an extensive set of computational tests are reported and discussed.


Transportation Stochastic demands Fixed handling costs  Two-stage stochastic programming 

Mathematics Subject Classification (2010)

90B06 90B15 90C15 



This research has been partially supported by projects FQM-5849 (Junta de Andalucía\(\backslash \)FEDER) and MTM2010-19576-C02-01 (MICINN, Spain) and by the Portuguese Science Foundation—Centro de Investigação Operacional. The authors would like to express their gratitude to the anonymous referees for the constructive comments and suggestions given, which helped to improve the paper.


  1. Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms, and applications. Prentice Hall, LondonGoogle Scholar
  2. Barbarosoǧlu G, Arda Y (2004) A two-stage stochastic programming framework for transportation planning in disaster response. J Oper Res Soc 55:43–53CrossRefGoogle Scholar
  3. Birge JR (1982) The value of the stochastic solution in stochastic linear programs with fixed recourse. Math Prog 24:314–325CrossRefGoogle Scholar
  4. Birge JR, Louveaux F (1997) Introduction to stochastic programming. Springer, LondonGoogle Scholar
  5. Escudero LF (2009) On a mixture of the fix-and-relax coordination and lagrangian substitution schemes for multistage stochastic mixed integer programming. TOP 17:5–29CrossRefGoogle Scholar
  6. Escudero LF, Garín A, Merino M, Prezé G (2007) The value of the stochastic solution in multistage problems. TOP 15:48–64CrossRefGoogle Scholar
  7. Escudero LF, Garín MA, Pérez G, Unzueta A (2012) Lagrangian decomposition for large-scale two-stage stochastic mixed 0–1 problems. TOP 20:347–374CrossRefGoogle Scholar
  8. França PM, Luna HPL (1982) Solving stochastic transportation–location problems by generalized Benders decomposition. Transp Sci 16(2):113–126CrossRefGoogle Scholar
  9. Grieco S, Semeraro U, Tolio T (2001) A review of different approaches to the fms loading problem. Int J Flexible Manuf Syst 13(4):361–384CrossRefGoogle Scholar
  10. Holmberg K (1995) Efficient decomposition and linearization methods for the stochastic transportation problem. Comput Optim Appl 4:293–316CrossRefGoogle Scholar
  11. Holmberg K, Jörnsten KO (1984) Cross decomposition applied to the stochastic transportation problem. Eur J Oper Res 17(3)Google Scholar
  12. Holmberg K, Tuy H (1999) A production–transportation problem with stochastic demand and concave production costs. Math Prog 85:157–179CrossRefGoogle Scholar
  13. LeBlanc LJ (1977) A heuristic approach for large scale discrete stochastic transportation–location problems. Computers Math Appl 3:87–94CrossRefGoogle Scholar
  14. Li ACY, Nozick L, Xu N, Davidson R (2012) Shelter location and transportation planning under hurricane conditions. Transp Res Part E: Logistics Transp Rev 48:715–729CrossRefGoogle Scholar
  15. Lium A-G, Crainic TG, Wallace SW (2009) A study of demand stochasticity in service network design. Transp Sci 43:144–157CrossRefGoogle Scholar
  16. Max Shen Z-J, Coullard C, Daskin MS (2003) Joint location–inventory model. Transp Sci 37(1):40–55CrossRefGoogle Scholar
  17. Qi L (1985) Forest iteration method for stochastic transportation problem. Math Prog Study 25:142–163CrossRefGoogle Scholar
  18. Thapalia BK, Crainic TG, Kaut M, Wallace SW (2012a) Single-commodity network design with random edge capacities. Eur J Oper Res 220:394–403CrossRefGoogle Scholar
  19. Thapalia BK, Crainic TG, Kaut M, Wallace SW (2012b) Single-commodity stochastic network design with multiple sources and sinks. INFOR 49:193–211Google Scholar
  20. Tsai M-T, Saphores J-D, Regan A (2011) Valuation of freight transportation contracts under uncertainty. Transp Res Part E: Logistics Transp Rev 47:920–932CrossRefGoogle Scholar
  21. Williams AC (1963) A stochastic transportation problem. Oper Res 11(5):759–770CrossRefGoogle Scholar
  22. Xu N, Nozick L (2009) Modeling supplier selection and the use of option contracts for global supply chain design. Computers Oper Res 36:2786–2800CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2014

Authors and Affiliations

  • Yolanda Hinojosa
    • 1
  • Justo Puerto
    • 2
  • Francisco Saldanha-da-Gama
    • 3
    Email author
  1. 1.Departamento de Economía Aplicada IUniversidad de SevillaSevillaSpain
  2. 2.Departamento Estadística e Investigación Operativa, Facultad de MatemáticasUniversidad de SevillaSevillaSpain
  3. 3.DEIO-CIO, Faculdade de CiênciasUniversidade de LisboaLisboaPortugal

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