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

Abstract

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.

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Notes

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

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Acknowledgements

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).

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Correspondence to Angelo Aliano Filho.

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Appendix

Appendix

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
figure6

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

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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). https://doi.org/10.1007/s10479-019-03468-9

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Keywords

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