**Crop Planning Models with Symmetric Risk Measures**

**Marius RĂDULESCU ^{1}, Constanţa Zoie RĂDULESCU^{2}**

**Institute of Mathematical Statistics and Applied Mathematics,**

^{1 }Casa Academiei Române,

13, 13 Septembrie Avenue, Bucharest 5, RO-050711, Romania mradulescu.csmro@yahoo.com

^{2 }I C I Bucharest

(National Institute for R & D in Informatics)

8-10 Averescu Blvd.

011455 Bucharest 1, Romania

radulescucz@yahoo.com

**Abstract**: In this paper the financial risk of crop plans is measured by two symmetric risk measures: variance and mean absolute deviation of the return. Several crop planning models with symmetric risk measures, based on the financial portfolio theory, are formulated. Among them minimum risk and maximum expected return models are of interest. The decision variables are the land areas allocated to crops. The models belong to mathematical programming with continuous variables. Some numerical examples for the minimum financial risk model are studied and efficient frontiers of the models are displayed.

**Keywords**: crop planning, symmetric risk measures, portfolio theory, efficient frontier, decision support.

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**CITE THIS PAPER AS**:

Marius RĂDULESCU, Constanţa Zoie RĂDULESCU, **Crop Planning Models with Symmetric Risk Measures**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 23 (4), pp. 333-340, 2014.

**Introduction**

Agriculture today enjoys a special attention in all the countries since it contributes to their economic development. In today’s world most economically developed countries are also the largest producers and exporters of basic agricultural products. Due to the cyclical nature of agricultural product prices, farmers are often faced with management challenges. When crop prices are low and margins are small, crop planning is increasingly important as farm managers strive to maximize net farm income.

The decisions connected to what crops to grow and the land areas allocated to each crop to plant, are complex management decisions. Each year, farm managers go through a process of deciding what crops to grow on each field.

When farmers are making crop plan decisions the primary consideration is economics. However once they have determined the crops that will provide the highest net returns they will often consider rotations, weed problems, herbicide residues and various other factors.

One of the main objectives when a crop plan is made is to find the combination of crops that will provide the maximum expected net returns per hectare. Another main objective is connected to the minimization of farmer income variability. It is important to consider the crops with higher expected net return in terms of risk and probability of achieving the highest level of net return. Many of the specialty crops are higher risk crops and usually require greater managerial input and marketing skills in order to achieve this higher net return. When making long term planning, decisions farmers should calculate, for all crops that are considering growing, the total revenue, total expenses and return, over total expenses.

Prospective growers must know how to use risk management strategies in order to select the crops that best suit their needs. One of the most popular approaches to managing risk is to reduce risk exposure through diversification. The uncertainty from the agriculture problems is modeled with the help of probability theory. Many of the practical problems that occur in agriculture are modelled with stochastic programming. Most of the problems that appear in crop planning have multiple objectives. In practice, in the process of mathematical modeling, one cannot take into account all the factors that have an impact to agricultural production. The number of such factors is large and the growth of their number determines the rapid growth of the complexity of the models. An important mathematical instrument which was successfully applied to modeling the problems from agriculture is portfolio theory. The above mentioned theory is widely used in to determine investment strategies under uncertainty. It shows how investments in different assets can be combined in a portfolio with a lower risk.

The application of portfolio theory for finding an optimal allocation of crops to agricultural land is popular in the literature. The first application of the portfolio theory to crop planning goes back to Freund [6]. In Hardaker [8], Hazell [9], Hazell and Norton [10] and Blank [2]-[4] were presented or applied various variants of portfolio theory to the land allocation decisions. In Collender [5], Romero [17]-[19], were studied several models for resources allocation in agriculture that are taking into account specific risks. In Werners et al. [21] portfolio theory was used for the evaluation of agricultural land use, as an adaptation to climate risk in the Hungarian Tisza River Basin. The main climate related risk in the Tisza is the frequent occurrence of floods and droughts.

In Barkley et al [1] portfolio theory was used to find the optimal, yield-maximizing and risk-minimizing combination of wheat varieties in Kansas.

In Nalley [12] portfolio theory is applied to wheat varietal selection decisions in order to find risk-minimizing outcomes. The selection of wheat varieties, through portfolio theory, offers producers in low-income countries the potential to increase yield or decrease yield variability. Farmers in low-income countries often value yield stability as much as yield potential. They frequently have a choice of several wheat varieties to sow and must evaluate the tradeoff between yield, mean and variance. Using location-specific empirical data, portfolio theory can provide producers in low-income countries a tool for developing a recommended varieties portfolio given a desired risk-aversion level.

Usually, in the crop planning models, the risk is measured by variance. In this case the risk becomes a quadratic function. The minimum risk model is a QP model. In Hazell [9] was developed a linear alternative to the quadratic model that uses variance as a measure for risk. The Hazell’s model was called the MOTAD (Minimization Of Total Absolute Deviation) model. The computational advantage of the MOTAD is that it can be transformed in an equivalent linear programming model. The MOTAD model is extensively used in international studies in recent decades Hardaker et al [8]; Zia [22], McCarl [11], Zimet and Spreen [23], Vadnere and Padney [20].

For other references regarding applications of portfolio theory to agriculture see Radulescu [13], [14] and [16].

Our research is focused on the formulation of several original models for crop planning, based on financial portfolio theory. To evaluate risk of the net return we use two symmetric risk measures: the variance and the mean absolute deviation. We formulate two minimum financial risk models and two maximum expected net return models. In contrast to the models developed in Radulescu and Radulescu [14] and Radulescu et al. [16], which are binary and respectively mixed-binary, the models from the present paper are formulated in continuous variable. Our models can be embedded as modules in a decision support system for crop planning. Some numerical examples for the minimum financial risk model are studied and efficient frontiers of the models are displayed. The models are solved with QP and LP solvers from GAMS.

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