Skip to main content

Exact and heuristic methods to solve a bi-objective problem of sustainable cultivation


This work proposes a binary nonlinear bi-objective optimization model for the problem of planning the sustainable cultivation of crops. The solution to the problem is a planting schedule for crops to be cultivated in predefined plots, in order to minimize the possibility of pest proliferation and maximize the profit of this process. Biological constraints were also considered. Exact methods, based on the nonlinear model and on a linearization of that model were proposed to generate Pareto optimal solutions for the problem of sustainable cultivation, along with a metaheuristic approach for the problem based on a genetic algorithm and on constructive heuristics. The methods were tested using semi-randomly generated instances to simulate real situations. According to the experimental results, the exact methodologies performed favorably for small and medium size instances. The heuristic method was able to potentially determine Pareto optimal solutions of good quality, in a reduced computational time, even for high dimension instances. Therefore, the mathematical models and the methods proposed may support a powerful methodology for this complex decision-making problem.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    The number n must be a multiple of 2, because n / 2 needs to be integer.


  1. Alfandari, L., Lemalade, J.-L., Nagih, A., & Plateau, G. (2011). A mip flow model for crop-rotation planning in a context of forest sustainable development. Annals of Operations Research, 190(1), 149–164.

    Article  Google Scholar 

  2. Aliano Filho, A., de Oliveira Florentino, H., & Vaz Pato, M. (2014). Metaheuristics for a crop rotation problem. International Journal of Metaheuristics, 3(3), 199–222.

    Article  Google Scholar 

  3. Annetts, J., & Audsley, E. (2002). Multiple objective linear programming for environmental farm planning. Journal of the Operational Research Society, 53(9), 933–943.

    Article  Google Scholar 

  4. Boyabatlı, O., Nasiry, J., & Zhou, Y. (2019). Crop planning in sustainable agriculture: Dynamic farmland allocation in the presence of crop rotation benefits. Management Science, 65(5), 2060–2076.

    Google Scholar 

  5. Costa, A. M., dos Santos, L. M. R., Alem, D. J., & Santos, R. H. (2014). Sustainable vegetable crop supply problem with perishable stocks. Annals of Operations Research, 219(1), 265–283.

    Google Scholar 

  6. CPLEX. (2017). IBM ILOG CPLEX 12.8 Optimizer.

  7. de Oliveira Florentino, H., Irawan, C., Aliano, A. F., Jones, D. F., Cantane, D. R., Nervis, J. J., et al. (2018). A multiple objective methodology for sugarcane harvest management with varying maturation periods. Annals of Operations Research, 267(1–2), 153–177.

    Article  Google Scholar 

  8. Deb, K. (2001). Multi-objective optimization using evolutionary algorithms., Wiley-Interscience series in systems and optimization Chichester: Wiley.

    Google Scholar 

  9. Dogliotti, S., Van Ittersum, M., & Rossing, W. (2005). A method for exploring sustainable development options at farm scale: A case study for vegetable farms in south Uruguay. Agricultural Systems, 86(1), 29–51.

    Article  Google Scholar 

  10. Dury, J., Schaller, N., Garcia, F., Reynaud, A., & Bergez, J. E. (2012). Models to support cropping plan and crop rotation decisions. A review. Agronomy for Sustainable Development, 32(2), 567–580.

    Article  Google Scholar 

  11. Han, K.-H., & Kim, J.-H. (2000). Genetic quantum algorithm and its application to combinatorial optimization problem. In Proceedings of the 2000 Congress on Evolutionary Computation, 2000 (Vol. 2, pp. 1354–1360) IEEE.

  12. Havlin, J., Kissel, D., Maddux, L., Claassen, M., & Long, J. (1990). Crop rotation and tillage effects on soil organic carbon and nitrogen. Soil Science Society of America Journal, 54(2), 448–452.

    Article  Google Scholar 

  13. Hayashi, K. (2000). Multicriteria analysis for agricultural resource management: A critical survey and future perspectives. European Journal of Operational Research, 122(2), 486–500.

    Article  Google Scholar 

  14. Huang, I. B., Keisler, J., & Linkov, I. (2011). Multi-criteria decision analysis in environmental sciences: Ten years of applications and trends. Science of The Total Environment, 409(19), 3578–3594.

    Article  Google Scholar 

  15. Kumar, A., Sah, B., Singh, A. R., Deng, Y., He, X., Kumar, P., et al. (2017). A review of multi criteria decision making (MCDM) towards sustainable renewable energy development. Renewable and Sustainable Energy Reviews, 69, 596–609.

    Article  Google Scholar 

  16. MATLAB. (2017). version 7.10.0 (R2017a). The MathWorks Inc., Natick, Massachusetts.

  17. Miettinen, K. (2012). Nonlinear multiobjective optimization (Vol. 12). New York: Springer.

    Google Scholar 

  18. Nechi, S., Aouni, B., & Mrabet, Z. (2019). Managing sustainable development through goal programming model and satisfaction functions. Annals of Operations Research, 1–20.

  19. Parra, J. R. P. (2002). Controle Biológico no Brasil: Parasitóides e Predadores. São Paulo: Editora Manole Ltda.

    Google Scholar 

  20. Rossing, W., Hammer, G., et al. (2006). Exploring profit-sustainability trade-offs in cropping systems using evolutionary algorithms. Environmental Modelling & Software, 21(9), 1368–1374.

    Article  Google Scholar 

  21. Santos, L. M., Costa, A. M., Arenales, M. N., & Santos, R. H. S. (2010). Sustainable vegetable crop supply problem. European Journal of Operational Research, 204(3), 639–647.

    Article  Google Scholar 

  22. Santos, L. M., Michelon, P., Arenales, M. N., & Santos, R. H. S. (2011). Crop rotation scheduling with adjacency constraints. Annals of Operations Research, 190(1), 165–180.

    Article  Google Scholar 

  23. Santos, L. M., Munari, P., Costa, A. M., & Santos, R. H. (2015a). A branch-price-and-cut method for the vegetable crop rotation scheduling problem with minimal plot sizes. European Journal of Operational Research, 245(2), 581–590.

    Article  Google Scholar 

  24. Santos, L. M., Munari, P., Costa, A. M., & Santos, R. H. (2015b). A branch-price-and-cut method for the vegetable crop rotation scheduling problem with minimal plot sizes. European Journal of Operational Research, 245(2), 581–590.

    Article  Google Scholar 

  25. Sarker, R., & Ray, T. (2009). An improved evolutionary algorithm for solving multi-objective crop planning models. Computers and Electronics in Agriculture, 68(2), 191–199.

    Article  Google Scholar 

  26. Snapp, S., Swinton, S., Labarta, R., Mutch, D., Black, J., Leep, R., et al. (2005). Evaluating cover crops for benefits, costs and performance within cropping system niches. Agronomy Journal, 97(1), 322–332.

    Google Scholar 

  27. Tilman, D., Cassman, K. G., Matson, P. A., Naylor, R., & Polasky, S. (2002). Agricultural sustainability and intensive production practices. Nature, 418(6898), 671.

    Article  Google Scholar 

  28. West, T. O., & Post, W. M. (2002). Soil organic carbon sequestration rates by tillage and crop rotation. Soil Science Society of America Journal, 66(6), 1930–1946.

    Article  Google Scholar 

Download references


The authors thank the Brazilian institutions FAPESP (Projects n. 2014/01604-0 and 2014/04353-8), CNPq (n.302454/2016-0), FAPESP (n. 2013/07375-0). In addition, we thank the Federal Technological University of Paraná for its support to this research and the translation services provided. The research performed by Margarida Vaz Pato was supported by funding from Fundação para a Ciência e a Tecnologia, Portugal, under projects UID/MAT/04561/2013 and UID/Multi/00491/2013 and by the Research Fund of ISEG. Helenice de Oliveira Florentino was supported by funding from Conselho Nacional de Desenvolvimento Científico e Tecnológico (303267/2011-9).

Author information



Corresponding author

Correspondence to Angelo Aliano Filho.

Additional information

Publisher's Note

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



Table 9 presents the characteristics of all instances used.

Table 9 Characteristics of the instances: values for the parameters, number of variables and constraints for models (1)–(6) and (13)–(20)

Table 10 presents the values of the parameters used by the proposed GA to solve the problem.

Table 10 Parameters used for the implementation of the GA for all instances

The geometry and layout of plots to create the 27 instances are shown in Fig. 6. The area of each plot was fixed in 4 hectares. In Fig. 6 we illustrate a plantation area divided into 12 plots.

Table 11 presents data for 20 different crops, specifically for duration of the crop cultivation, botanical family that the plant belongs to, and profitability per hectare.

Fig. 6

Illustration of the disposition and dimension of the plots considered for the instances

The area of each plot j was fixed equal to \(area_j=4\) hectares for all j.

Table 11 Crop data: duration of the crop cultivation, family they belong to and profitability per hectare

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aliano Filho, A., de Oliveira Florentino, H., Pato, M.V. et al. Exact and heuristic methods to solve a bi-objective problem of sustainable cultivation. Ann Oper Res (2019).

Download citation


  • Multi-objective optimization
  • Genetic algorithm
  • Constructive heuristics and sustainability