, Volume 22, Issue 1, pp 343–362 | Cite as

Modeling target volume flows in forest harvest scheduling subject to maximum area restrictions

  • Isabel Martins
  • Mujing Ye
  • Miguel Constantino
  • Maria da Conceição Fonseca
  • Jorge Cadima
Original Paper


In forest harvest scheduling problems, one must decide which stands to harvest in each period during a planning horizon. A typical requirement in these problems is a steady flow of harvested timber, mainly to ensure that the industry is able to continue operating with similar levels of machine and labor utilizations. The integer programming approaches described use the so-called volume constraints to impose such a steady yield. These constraints do not directly impose a limit on the global deviation of the volume harvested over the planning horizon or use pre-defined target harvest levels. Addressing volume constraints generally increases the difficulty of solving the integer programming formulations, in particular those proposed for the area restriction model approach. In this paper, we present a new type of volume constraint as well as a multi-objective programming approach to achieve an even flow of timber. We compare the main basic approaches from a computational perspective. The new volume constraints seem to more explicitly control the global deviation of the harvested volume, while the multi-objective approach tends to provide the best profits for a given dispersion of the timber flow. Neither approach substantially changed the computational times involved.


Forest harvest scheduling Area restriction model Volume constraint Integer programming Multi-objective programming 

Mathematics Subject Classification

90B50 90C10 



This research was partially supported by Centro de Investigação Operacional (through the project POCTI/ISGL/152) and Centro de Estatística e Aplicações from Universidade de Lisboa. We wish to thank Andres Weintraub and José G. Borges (through the project PTDC/AGR-CFL/64146/2006) for providing some real test forest data.


  1. Bertomeu M, Diaz-Balteiro L, Giménez JC (2009) Forest management optimization in Eucalyptus plantations: a goal programming approach. Can J For Res 39:356–366 CrossRefGoogle Scholar
  2. Bettinger P, Sessions J, Johnson KN (1998) Ensuring the compatibility of aquatic habitat and commodity production goals in Eastern Oregon with a Tabu Search procedure. For Sci 44:96–112 Google Scholar
  3. Brumelle S, Granot D, Halme M, Vertinsky I (1998) A Tabu Search algorithm for finding good forest harvest schedules satisfying green-up constraints. Eur J Oper Res 106:408–424 CrossRefGoogle Scholar
  4. Buongiorno J, Gilless JK (2003) Decision methods for forest resource management. Academic Press, New York Google Scholar
  5. Caro F, Constantino M, Martins I, Weintraub A (2003) A 2-opt tabu search procedure for the multiperiod forest harvesting problem with adjacency, greenup, old growth, and even flow constraints. For Sci 49(5):738–751 Google Scholar
  6. Chalmet LG, Lemonidis L, Elzinga DJ (1986) An algorithm for the bi-criterion integer programming problem. Eur J Oper Res 25:292–300 CrossRefGoogle Scholar
  7. Clements SE, Dallain PL, Jamnick MS (1990) An operational spatially constrained harvest scheduling model. Can J For Res 20:1438–1447 CrossRefGoogle Scholar
  8. Cohon JL (1978) Multiobjective programming and planning. Academic Press, New York Google Scholar
  9. Constantino M, Martins I, Borges JG (2008) A new mixed integer programming model for harvest scheduling subject to maximum area restrictions. Oper Res 56(3):542–551 CrossRefGoogle Scholar
  10. Crowe K, Nelson J, Boyland M (2003) Solving the area-restricted harvest-scheduling model using the branch and bound algorithm. Can J For Res 33(9):1804–1814 CrossRefGoogle Scholar
  11. Davis LS, Johnson KN, Bettinger P, Howard T (2001) Forest management. McGraw-Hill, New York Google Scholar
  12. Diaz-Balteiro L, Romero C (2003) Forest management optimisation models when carbon captured is considered: a goal programming approach. For Ecol Manag 174:447–457 CrossRefGoogle Scholar
  13. Diestel R (2000) Graph theory. Graduate texts in mathematics. Springer, Berlin Google Scholar
  14. Diwekar U (2008) Introduction to applied optimization. Springer, Berlin CrossRefGoogle Scholar
  15. Falcão AO (1997) DUNAS—a growth model for the National Forest of Leiria. In: Empirical and process-based models for forest tree and stand growth simulation. September 97, Portugal Google Scholar
  16. Falcão AO, Borges JG (2002) Combining random and systematic search procedures for solving spatially constrained forest management scheduling models. For Sci 48(3):608–621 Google Scholar
  17. Geoffrion AM (1968) Proper efficiency and the theory of vector maximization. J Math Anal Appl 22:618–630 CrossRefGoogle Scholar
  18. Gómez T, Hernández M, Molina J, León MA, Aldana E, Caballero R (2011) A multiobjective model for forest planning with adjacency constraints. Ann Oper Res 190(1):75–92 CrossRefGoogle Scholar
  19. Goycoolea M, Murray AT, Barahona F, Epstein R, Weintraub A (2005) Harvest scheduling subject to maximum area restrictions: exploring exact approaches. Oper Res 53(3):90–500 CrossRefGoogle Scholar
  20. Goycoolea M, Murray AT, Vielma JP, Weintraub A (2009) Evaluating approaches for solving the area restricted model in harvest scheduling. For Sci 55(2):149–165 Google Scholar
  21. Gunn EA, Richards EW (2005) Solving the adjacency problem with stand-centered constraints. Can J For Res 35:832–842 CrossRefGoogle Scholar
  22. ILOG (2007) ILOG CPLEX 11.0—user’s manual Google Scholar
  23. Jamnick MS, Walters KR (1993) Spatial and temporal allocation of stratum-based harvest schedules. Can J For Res 23:402–413 CrossRefGoogle Scholar
  24. Martins I, Constantino M, Borges JG (1999) Forest management models with spatial structure constraints. Working Paper No 2/1999, CIO, Faculdade de Ciências de Lisboa Google Scholar
  25. Martins I, Constantino M, Borges JG (2005) A column generation approach for solving a non-temporal forest harvest model with spatial structure constraints. Eur J Oper Res 161(2):478–498 CrossRefGoogle Scholar
  26. McDill ME, Rebain SA, Braze J (2002) Harvest scheduling with area-based adjacency constraints. For Sci 48(4):631–642 Google Scholar
  27. Murray AT (1999) Spatial restrictions in harvest scheduling. For Sci 45(1):45–52 Google Scholar
  28. Murray AT, Weintraub A (2002) Scale and unit specification influences in harvest scheduling with maximum area restrictions. For Sci 48(4):779–789 Google Scholar
  29. Neter J, Kutner MH, Nachsteim CJ, Wasserman W (1996) Applied linear statistical models, 4th edn. Irwin, New York Google Scholar
  30. O’Hara AJ, Faaland BH, Bare BB (1989) Spatially constrained timber harvest scheduling. Can J For Res 19:715–724 CrossRefGoogle Scholar
  31. Roise JP (1990) Multicriteria nonlinear programming for optimal spatial allocation of stands. For Sci 36:487–501 Google Scholar
  32. Romero C, Rehman T (2003) Multiple criteria analysis for agricultural decisions. Elsevier, Amsterdam Google Scholar
  33. Ross T, Soland R (1980) A multicriteria approach to the location of public facilities. Eur J Oper Res 4:307–321 CrossRefGoogle Scholar
  34. Steuer RE (1989) Multiple criteria optimization: theory. Computation and application. Wiley, New York Google Scholar
  35. Vielma JP, Murray AT, Ryan DM, Weintraub A (2007) Improving computational capabilities for addressing volume constraints in forest harvest scheduling problems. Eur J Oper Res 176(2):1246–1264 CrossRefGoogle Scholar
  36. Ware GO, Clutter JL (1971) A mathematical programming system for the management of industrial forests. For Sci 17:428–445 Google Scholar
  37. Weintraub A, Cholaky A (1991) A hierarchical approach to forest planning. For Sci 37(2):439–460 Google Scholar
  38. Yoshimoto A, Brodie JD (1994) Comparative analysis of algorithms to generate adjacency constraints. Can J For Res 24:1277–1288 CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2012

Authors and Affiliations

  • Isabel Martins
    • 1
  • Mujing Ye
    • 2
  • Miguel Constantino
    • 3
  • Maria da Conceição Fonseca
    • 3
  • Jorge Cadima
    • 1
  1. 1.Departamento de Ciências e Engenharia de BiosistemasInstituto Superior de AgronomiaLisboaPortugal
  2. 2.Northwestern UniversityEvanstonUSA
  3. 3.Departamento de Estatística e Investigação Operacional, Faculdade de Ciências de LisboaCentro de Investigação OperacionalLisboaPortugal

Personalised recommendations