Advertisement

Computational Management Science

, Volume 15, Issue 3–4, pp 583–597 | Cite as

A fractional stochastic integer programming problem for reliability-to-stability ratio in forest harvesting

  • Miguel A. LejeuneEmail author
  • Janne Kettunen
Original Paper

Abstract

We propose a new fractional stochastic integer programming model for forestry revenue management. The model takes into account the main sources of uncertainties—wood prices and tree growth—and maximizes a reliability-to-stability revenue ratio that reflects two major goals pursued by forest owners. The model includes a joint chance constraint with multirow random technology matrix to account for reliability and a joint integrated chance constraint to account for stability. We propose a reformulation framework to obtain an equivalent mixed-integer linear programming formulation amenable to a numerical solution. We use a Boolean modeling framework to reformulate the chance constraint and a series of linearization techniques to handle the nonlinearities due to the joint integrated chance constraint, the fractional objective function, and the bilinear terms. The computational study attests that the reformulation of the model can handle large number of scenarios and can be solved efficiently for sizable forest harvesting problems.

Keywords

Stochastic programming Joint probabilistic constraint Integrated chance constraint Forestry management Fractional programming 

Notes

Acknowledgements

We are grateful to Mikko Kurttila and the Finnish Forest Research Institute for providing the data used in this paper and for providing insightful suggestions about the models and the analysis of their results. M. Lejeune was partially supported by the Office of Naval Research, Grant #N000141712420.

References

  1. Ahmed S, Luedtke J, Song Y, Xie WWK (2017) Nonanticipative duality, relaxations, and formulations for chance-constrained stochastic programs. Math Program 162(1–2):51–81CrossRefGoogle Scholar
  2. Alonso-Ayuso A, Escudero LF, Guignard M, Quiteros M, Weintraub A (2011) Forestry management under uncertainty. Ann Oper Res 190(1):17–39CrossRefGoogle Scholar
  3. Bjørndal T, Herrero I, Newman A, Romero C, Weintraub A (2012) Operations research in the natural resource industry. Int Trans Oper Res 19(1–2):39–62CrossRefGoogle Scholar
  4. Boros E, Hammer PL, Ibaraki T, Kogan A (1997) Logical analysis of numerical data. Math Program 79(1–3):163–190Google Scholar
  5. Bullard SH (2001) How to evaluate the financial maturity of timber. For Landowner 60(3):36–38Google Scholar
  6. Eyvindson KJ, Petty AD, Kangas AS (2017) Determining the appropriate timing of the next forest inventory: incorporating forest owner risk preferences and the uncertainty of forest data quality. Ann For Sci 74(2):1–10Google Scholar
  7. Favada IM, Karppinen H, Kuuluvainen J, Mikkola J, Stavness C (2009) Effects of timber prices, ownership objectives, and owner characteristics on timber supply. For Sci 55(6):512–523Google Scholar
  8. Ferreira L, Constantino M, Borges JG (2014) A stochastic approach to optimize maritime pine (pinus pinaster ait.) stand management scheduling under fire risk. An application in Portugal. Ann Oper Res 219(1):359–377CrossRefGoogle Scholar
  9. Food and Agriculture Organization of the United Nations (2014) Trends and status of forest products and services. http://www.fao.org/docrep/w4345e/w4345e05.htm. Accessed 26 Sept (2017)
  10. Gassmann HI (1989) Optimal harvest of a forest in the presence of uncertainty. Can J For Res 19(10):1267–1274CrossRefGoogle Scholar
  11. Gunn EA, Rai AK (1987) Modelling and decomposition for planning long-term forest harvesting in an integrated industry structure. Can J For Res 17(12):1507–1518CrossRefGoogle Scholar
  12. Haneveld WKK (1986) Duality in stochastic linear and dynamic programming. Volume 274 of lecture notes in economics and mathematical systems. Springer-Verlag, BerlinCrossRefGoogle Scholar
  13. Haneveld WKK, van der Vlerk MH (2006) Integrated chance constraints: reduced forms and an algorithm. Comput Manag Sci 3(4):245–269CrossRefGoogle Scholar
  14. Haneveld WKK, Streutker MH, van der Vlerk MH (2010) An ALM model for pension funds using integrated chance constraints. Ann Oper Res 177(1):47–62CrossRefGoogle Scholar
  15. Häyrinen L, Mattila O, Berghäll S, Toppinen A (2015) Forest owners socio-demographic characteristics as predictors of customer value: evidence from Finland. Small-Scale For 14(1):19–37CrossRefGoogle Scholar
  16. Heikkinen V-P (2003) Timber harvesting as a part of the portfolio management: a multiperiod stochastic optimisation approach. Manage Sci 49(1):131–142CrossRefGoogle Scholar
  17. Innofor (2017) Everything starts from a forest plan (translated from Finnish). http://www.ostammepuuta.fi/metsanhoidon-palvelut/metsasuunnittelu/. Accessed 26 Sept 2017
  18. Jacobson M (2008) To cut or not to cut: tree value and deciding when to harvest timber. Technical report, http://extension.psu.edu/natural-resources/forests/finance/forest-tax-info/publications/forest-finance-8-to-cut-or-not-cut-tree-value-and-deciding-when-to-harvest-timber. Accessed 26 Sept 2017
  19. Kataoka S (1963) A stochastic programming model. Econometrica 31(1–2):181–196CrossRefGoogle Scholar
  20. Kogan A, Lejeune MA (2014) Threshold boolean form for joint probabilistic constraints with random technology matrix. Math Program 147(1):391–427CrossRefGoogle Scholar
  21. Lejeune MA (2012a) Pattern-based modeling and solution of probabilistically constrained optimization problems. Oper Res 60(6):1356–1372CrossRefGoogle Scholar
  22. Lejeune MA (2012b) Pattern definition of the \(p\)-efficiency concept. Ann Oper Res 200(1):23–36CrossRefGoogle Scholar
  23. Lejeune MA, Kettunen J (2017) Managing reliability and stability risks in forest harvesting. Manuf Serv Oper Manag 19(4):620–638CrossRefGoogle Scholar
  24. Lejeune MA, Margot F (2016) Solving chance constrained problems with random technology matrix and stochastic quadratic inequalities. Oper Res 64(4):939–957CrossRefGoogle Scholar
  25. Lejeune MA, Shen S (2016) Multi-objective probabilistically constrained programs with variable risk: models for multi-portfolio financial optimization. Eur J Oper Res 252(2):522–539CrossRefGoogle Scholar
  26. Marques AS, Audy JF, D’Amours S, Rönnqvist M (2014) Tactical and operational harvest planning. In: Borges JG, Dias-Balteiro L, McDill ME, Rodriguez LCE (eds) Theoretical foundations and applications. Springer, New York, pp 239–267Google Scholar
  27. Martell DL (1980) The optimal rotation of a flammable forest stand. Can J For Res 10(1):30–34CrossRefGoogle Scholar
  28. McCormick GP (1976) Computability of global solutions to factorable nonconvex programs: part I. convex underestimating problems. Math Program 10(1):147–175CrossRefGoogle Scholar
  29. Mosquera J, Henig MI, Weintraub A (2011) Design of insurance contracts using stochastic programming in forestry planning. Ann Oper Res 190(1):117–130CrossRefGoogle Scholar
  30. Pasalodos-Tato M, Mäkinen A, Garcia-Gonzalo J, Borges JG, Lämås T, Eriksson LO (2013) Assessing uncertainty and risk in forest planning and decision support systems: review of classical methods and introduction of innovative approaches. For Syst 22(2):282–303Google Scholar
  31. Piazza A, Pagnoncelli BK (2014) The optimal harvesting problem under price uncertainty. Ann Oper Res 217(1):425–445CrossRefGoogle Scholar
  32. Richards EW, Gunn EA (2003) Tabu search design for difficult forest management optimization problems. Can J For Res 33(6):1126–1133CrossRefGoogle Scholar
  33. Rönnqvist M (2003) Optimization in forestry. Math Program 97(1–2):267–284CrossRefGoogle Scholar
  34. Ruszczyński A (2002) Probabilistic programming with discrete distribution and precedence constrained knapsack polyhedra. Math Program 93(2):195–215CrossRefGoogle Scholar
  35. Tahvonen O, Kallio M (2006) Optimal harvesting of forest age classes under price uncertainty and risk aversion. Nat Resour Model 19(4):557–585CrossRefGoogle Scholar
  36. Tanner MW, Ntaimo L (2010) IIS branch-and-cut for joint chance-constrained stochastic programs and application to optimal vaccine allocation. Eur J Oper Res 207(1):290–296CrossRefGoogle Scholar
  37. Weintraub A, Abramovich A (1995) Analysis of uncertainty of future timber yields in forest management. For Sci 41(2):217–234Google Scholar
  38. Weintraub A, Wets RJ-B (2014) Harvesting management: generating wood-prices scenarios. Available from www.math.ucdavis.edu/~rjbw/mypage/Stochastic Optimization files/WntW13.pdf

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.George Washington UniversityWashingtonUSA

Personalised recommendations