An Algorithm for Optimization of Cereal Output

Abstract

Algeria is rich country, it uses fertilizers to improve its cereal production as it is usually known. However, Algerian farmers use fertilizers in a bad manner and in a quite random way. This causes both ground, water pollution, and a loss of output.In the present paper we study a modelization of the evolution of the cereal output production controlled by means of adding fertilizers. We set an optimal control problem where we aim at maximizing the cereal output and meanwhile minimizing pollution effects. We solve theoretically this optimal control problem with the Pontryagin Maximum Principle and then provide different numerical methods to implement it: indirect methods based on the shooting method and direct methods based on a combination of AMPL with IPOPT. Mathematics Subject Classification (2010) 49K15 · 49M25 · 49J15 Statement of the ProblemIn a cereal field, consider x(t), t ∈ [0, T ] the rate of pollution at time t . In the absence of fertilizers, the rate of pollution x(t) is naturally (slowly) decreasing. Let α denote the natural decreasing rate. We assume that, in the absence of fertilizers, x(t) is evolving accordinġAdding fertilizers to the cereal field is a usual way in order to increase the cereal output, however it causes pollution. Denoting by u(t), t ∈ [0, T ], the rate of fertilizers, we assume that the evolution of the pollution rate x(t) is given bẏ(1) N. Moussouni ( ) · M. Aidene

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“…This problem is inspired by a model used in [9], where the authors formulated a model without presence of locusts. They calculated the quantities of fertilizers to put in cereal field to get a better production.…”

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“…This work is structured as follows: after a brief introduction, in section 2, we defined the model used and explained the importance of each of its compartments. Section 3 is devoted to the theoretical resolution of the modeled problem, using the Pontryagin maximum principle [4][5][6][7][8]. The numerical resolution is evoked in section 4, we solved the considered problem with two numerical methods, namely the direct method (Euler discretization ) and an indirect method (shooting method).…”

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