The evaluate stability and determine cropping patterns in agricultural systems by using multigoal mathematical programming

Stability of soil and water resources in agriculture, above all depends on the type of resource utilization and cropping pattern. Nowadays, Maximization of factors such as income, job opportunities and minimization costs as the economic and social aspects along with the limitations of arable land and water is considered. One of the most important questions is the issue of sustainability, how to evaluate, measure and analyze sustainability. One of the methods and models to assess the stability is mathematical programming approaches. Application of these models for agricultural planning has a long history and a wide range. But using a particular type of this models to assess the sustainability, means fractional models is a new topic. Lara and Stancu-Minasian (1999) examined the theoretical aspects of this model in a research titled as “Fractional programming: A tool for the assessment of sustainability”. As mentioned above, there are three main objectives for sustainable crop pattern, maximization income and job opportunities and minimization costs (In this case water consumption). Using fractional programming provides the possibility to make three goals in two goals which include maximization of “the net profit/water consumption” and “making job opportunity/water consumption”. In this way, the number of solutions is lower and therefore the decision-making process is easier. Fractional programming for planning and optimization cropping pattern has not been done yet. This paper is being studied to evaluate the stability in the agricultural systems of Bajestan plain in the Razavi Khorasan province; and also determine the best cropping pattern according to that. There are about 10,000 hectares of agricultural lands in the plains Bajestan. These lands are cultivated in seven major crops which include barley, saffron, wheat, pistachios, cotton, pomegranate and melon. Now the area under cultivation of these crops is 2250, 1630, 1500, 1400, 1200, 1350 and 670 hectares, respectively. In this region, about 70% of the working population are employed in agriculture.To verify the effectiveness of the fractional models first of all, three mono goal linear models were prepared. The goal of each of these three models was maximization income and maximization job opportunities and minimization water consumption. By solving those models, three cropping patterns A1, B1 and C1 was obtained. For combination the triple dimensions of social, economic, and environmental, two relative criteria have been explained as the indications to evaluate system stability. Thus, In the next step, by dividing each of the two functions (maximization income and maximization job opportunities) to water consumption, two new objective function were obtained in the form of fractional model. In fact, this two models represent two sustainability index. By solving those models, two cropping patterns A2 and B2 was obtained. Finally, an ideal fractional/mathematical programing model was prepared in which the last two functions were optimized. Optimization in relationships “the net profit/water consumption” and “making job opportunity/water consumption” leads to new copping pattern (C2).In optimized cropping pattern the area of barley, saffron, wheat, pistachios, cotton, pomegranate and melon obtained 111.4, 1608, 110.8, 7202.3, 103.1, 404.8 and 159.5 hectares, respectively. With the aim of maximizing the net profit, two relative criteria (“the net profit/water consumption” and “making job opportunity/water consumption”) in fractional model to the linear increase 17.3% and 19.0%, respectively. With the aim of maximizing “making job opportunity”, two relative criteria in fractional model to the linear increase 25.2% and 22.1%, respectively. In multi-goals programing approach, two relative criteria in fractional model to the linear increase 32.4% and 42.5%, respectively. Overall, the ratio of income to water consumption in three cropping pattern of linear programming (A1, B1 and C1) is lower than three cropping pattern of fractional programming (A2, B2 and C2). The values are respectively 17.8%, 26.7% and 33.3%. Also, the ratio of making job opportunity to water consumption in three cropping pattern of linear programming (A1, B1 and C1) is lower than three cropping pattern of fractional programming (A2, B2 and C2). The values are respectively 18.9%, 22.1% and 42.5%. It can be concluded that fractional models in terms of both sustainability index are superior to the linear models. These results are consistent with previous results and confirm it, such as Castrodeza et al. (2005), Maros et al. (2009), Hu et al. (2010) and Sabaghi et al.