Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Liner ship bunkering and sailing speed planning with uncertain demand

  • 16 Accesses


Liner shipping is an important branch of maritime transportation. As bunker fuel consumption causes high operating cost and harmful gas emissions, bunker fuel management is a great challenge and a hot research topic in liner shipping. Bunker charging can be achieved at ports with diverse prices, and it is recognized that appropriately managing bunker fuel and sailing speed can improve liner shipping performance and reduce environmental pollution. Most existing works assume that the container demand is deterministic. However, in practice, it is usually difficult to exactly estimate the volume of containers to be shipped due to various factors. This paper studies a liner ship bunkering and speed optimization problem under uncertain container demand. For the problem, a two-stage stochastic and non-linear programming formulation is proposed. To split the complexity of the problem, the complicated bunker consumption function is approximated by piecewise linear ones. To solve the problem, a classic sample average approximation (SAA) method, and the SAA based on scenario reduction, and an L-shaped method are developed and compared. Numerical results show that the L-shaped method outperforms the two SAA methods, in terms of solution quality and computational time.

This is a preview of subscription content, log in to check access.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. Bertsimas D, Gupta V, Kallus N (2017) Robust sample average approximation. Math Prog 171:217–282

  2. Birge JR, Louveaux F (2011) Introduction to stochastic programming. Springer, New York

  3. Corbett JJ, Wang H, Winebrake JJ (2009) The effectiveness and costs of speed reductions on emissions from international shipping. Transp Res Part D 14(8):593–598

  4. Dong JX, Lee CY, Song DP (2015) Joint service capacity planning and dynamic container routing in shipping network with uncertain demands. Transp Res Part D 78:404–421

  5. Du Y, Chen Q, Quan X, Long L, Fung RY (2011) Berth allocation considering fuel consumption and vessel emissions. Transp Res Part E 47(6):1021–1037

  6. Fagerholt K, Laporte G, Norstad I (2010) Reducing fuel emissions by optimizing speed on shipping routes. J Oper Res Soc 61(3):523–529

  7. Feng Y, Ryan SM (2016) Solution sensitivity-based scenario reduction for stochastic unit commitment. Comput Manag Sci 13(1):29–62

  8. Gabriel SA, Zhuang J, Egging R (2009) Solving stochastic complementarity problems in energy market modeling using scenario reduction. Eur J Oper Res 197(3):1028–1040

  9. Gelareh S, Meng Q (2010) A novel modeling approach for the fleet deployment problem within a short-term planning horizon. Transp Res Part E 46(1):76–89

  10. Heitsch H, Römisch W (2003) Scenario reduction algorithms in stochastic programming. Comput Optim Appl 24(2–3):187–206

  11. Heitsch H, Römisch W (2007) A note on scenario reduction for two-stage stochastic programs. Oper Res Lett 35(6):731–738

  12. Heitsch H, Römisch W (2009) Scenario tree modeling for multistage stochastic programs. Math Program 118(2):371–406

  13. Hvattum LM, Norstad I, Fagerholt K, Laporte G (2013) Analysis of an exact algorithm for the vessel speed optimization problem. Networks 62(2):132–135

  14. Kim HJ, Chang YT, Kim KT, Kim HJ (2012) An epsilon-optimal algorithm considering greenhouse gas emissions for the management of a ship’s bunker fuel. Transp Res Part D 17(2):97–103

  15. Kleywegt AJ, Nori VS, Savelsbergh MWP (2002) The stochastic inventory routing problem with direct deliveries. Transp Sci 36(1):94–118

  16. Laporte G, Louveaux FV (1993) The integer L-shaped method for stochastic integer programs with complete recourse. Oper Res Lett 13(3):133–142

  17. Laporte G, Louveaux FV, Van Hamme L (2002) An integer l-shaped algorithm for the capacitated vehicle routing problem with stochastic demands. Oper Res 50(3):415–423

  18. Lee TC, Wu CH, Lee PTW (2011) Impacts of the ecfa on seaborne trade volume and policy development for shipping and port industry in taiwan. Marit Policy Manag 38(2):169–189

  19. Lei C, Lin WH, Miao L (2014) A multicut L-shaped based algorithm to solve a stochastic programming model for the mobile facility routing and scheduling problem. Eur J Oper Res 238(3):699–710

  20. Liste O (2007) A generic stochastic model for supply-and-return network design. Comput Oper Res 34(2):417–442

  21. Liu M, Liu X, Zhang E, Chu F, Chu C (2018) Scenario-based heuristic to two-stage stochastic program for the parallel machine scheloc problem. Int J Prod Res.

  22. Meng Q, Wang T, Wang S (2012) Short-term liner ship fleet planning with container transshipment and uncertain container shipment demand. Eur J Oper Res 223(1):96–105

  23. Norstad I, Fagerholt K, Laporte G (2011) Tramp ship routing and scheduling with speed optimization. Transp Res Part C 19(5):853–865

  24. Notteboom TE, Vernimmen B (2009) The effect of high fuel costs on liner service configuration in container shipping. J Transp Geogr 17(5):325–337

  25. Pagnoncelli BK, Ahmed S, Shapiro A (2009) Sample average approximation method for chance constrained programming: Theory and applications. J Optim Theory Appl 142(2):399–416

  26. Psaraftis HN (2012) Market-based measures for greenhouse gas emissions from ships: a review. WMU J Marit Affairs 11(2):211–232

  27. Psaraftis HN, Kontovas CA (2013) Speed models for energy-efficient maritime transportation: a taxonomy and survey. Transp Res Part C 26:331–351

  28. Rachev ST, Römisch W (2002) Quantitative stability in stochastic programming: the method of probability metrics. Math Oper Res 27(4):792–818

  29. Ronen D (1982) The effect of oil price on the optimal speed of ships. J Oper Res Soc 33(11):1035–1040

  30. Ronen D (2011) The effect of oil price on containership speed and fleet size. J Oper Res Soc 62(1):211–216

  31. Rubasheuski U, Oppen J, Woodruff DL (2014) Multi-stage scenario generation by the combined moment matching and scenario reduction method. Oper Res Lett 42(5):374–377

  32. Trukhanov S, Ntaimo L, Schaefer A (2010) Adaptive multicut aggregation for two-stage stochastic linear programs with recourse. Eur J Oper Res 206(2):395–406

  33. UNCTAD (2017). Review of maritime transportation 2017. United Nations Conference on Trade and Development.

  34. Valentine VF, Benamara H, Hoffmann J (2013) Maritime transport and international seaborne trade. Marit Policy Manag 40(3):226–242

  35. Verweij B, Ahmed S, Kleywegt AJ, Nemhauser G, Shapiro A (2003) The sample average approximation method applied to stochastic routing problems: a computational study. Comput Optim Appl 24(2–3):289–333

  36. Wang S (2013) Essential elements in tactical planning models for container liner shipping. Transp Res Part B 54:84–99

  37. Wang S (2016) Fundamental properties and pseudo-polynomial-time algorithm for network containership sailing speed optimization. Eur J Oper Res 250(1):46–55

  38. Wang S, Meng Q (2012a) Liner ship route schedule design with sea contingency time and port time uncertainty. Transp Res Part B 46(5):615–633

  39. Wang S, Meng Q (2012b) Robust schedule design for liner shipping services. Transp Res Part E 48(6):1093–1106

  40. Wang S, Meng Q (2012c) Sailing speed optimization for container ships in a liner shipping network. Transp Res Part E 48(3):701–714

  41. Wang S, Meng Q (2015) Robust bunker management for liner shipping networks. Eur J Oper Res 243(3):789–797

  42. Wang S, Meng Q (2017) Container liner fleet deployment: a systematic overview. Transp Res Part C 77:389–404

  43. Wang S, Meng Q, Liu Z (2013a) Bunker consumption optimization methods in shipping: a critical review and extensions. Transp Res Part E 53:49–62

  44. Wang S, Meng Q, Liu Z (2013b) Containership scheduling with transit-time-sensitive container shipment demand. Transp Res Part B 54:68–83

  45. Wang S, Meng Q, Liu Z (2013c) A note on “berth allocation considering fuel consumption and vessel emissions”. Transp Res Part E 49(1):48–54

  46. Yao Z, Ng SH, Lee LH (2012) A study on bunker fuel management for the shipping liner services. Comput Oper Res 39(5):1160–1172

Download references


The authors would like to thank the anonymous referees for their constructive comments. This work was supported by the National Natural Science Foundation of China (NSFC) under Grants 71531011, 71771048, and 71571134. This work was also supported by the Fundamental Research Funds for the Central Universities.

Author information

Correspondence to Xin Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Hector Cancela.



The scenario reduction technique introduced by Heitsch and Römisch (2003), is presented in Algorithm 2, where \(p_{\omega }\) denotes the probability of scenario \(\omega \in \Omega \).


Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, M., Liu, X., Chu, F. et al. Liner ship bunkering and sailing speed planning with uncertain demand. Comp. Appl. Math. 39, 22 (2020).

Download citation


  • Sustainable liner shipping
  • Bunker consumption
  • Sailing speed
  • Uncertain container demand
  • Two-stage stochastic program

Mathematics Subject Classification

  • 90-08