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is defined as the overall monetary penalties incurred paid by a farmer. It is a nonsmooth
function of the crop plan. The multiobjective model for crop planning aims to minimize the
environmental risk, to maximize the expected return and to minimize the financial risk in the
presence of a set of constraints such as: the demand constraints, the budget constraints and
the environmental constraints. Starting from the multiobjective model for crop planning weformulate several single objective optimization problems: the minimum environmental risk
problem, the maximum expected return problem and the minimum financial risk problem.
We prove that the minimum environmental risk problem is equivalent to a mixed integer
problem with a linear objective function and linear and quadratic constraints. A numerical
case is analyzed for the minimum environmental risk problem. In the next section several
methods for diversification of agricultural production are presented. Our model covers the
first two types of production diversification. The methods developed in this paper can be
easily extended in order to also include the third type of diversification that is temporal
diversification.
2 Diversification of agricultural production
There are several definitions: see Ellis (1998, 2000). Farmers use three types of production
diversification. The most common type is diversification across products. This is a strategy
derived from portfolio theory developed in the stock market. This strategy can be applied
by any farmer with some knowledge of how to cultivate more than one crop, including
farmers with small and/or contiguous parcels of land. The goal of this type of diversification
is to reduce variance in sale revenues by participating in more than one product market. To
be successful, the product markets must have low or negative levels of correlation in their
prices.
The second type of diversification, across locations, has also been well known for some
time (Goland 1993), but practiced less often because it requires operating two or more
parcels that are geographically separated, a requirement which could be infeasible for some
farmers (Nartea and Bany 1994). Under this spatial diversification, a farmer must scatter
crop production across locations sufficiently far apart to have low levels of correlation in
their weather extremes. Thus, because the focus is on reducing yield variance, this strategy
can be applied by farmers specialized in growing one single crop.
Finally, cultivar diversity is a form of temporal diversification, but it incorporates aspectsof each of the other two diversification strategies. The common goal of cultivar diversity is
to have portions of total acreage (either contiguous or scattered) reach the harvest stage at
different times of the year (Park and Florkowski 2003). By selecting cultivars of a single
crop that are not highly correlated in their growth schedules, farmers can both:
(a) reduce average yield variability by reducing weather risk exposure (a feature of geo-
graphical diversification), and
(b) raise average price received and/or lower price variance by being able to sell output in
more than one market season (similar to product diversification).
The practice of crop diversification complicates both production and marketing but it can
increase profits. Diversification is a favorite risk-management tool of many farmers.
For example, a survey in California found most farmers use some type of diversification
as a risk management strategy, while few producers use the available financial risk man-
agement tools (Blank et al. 1997). That study reported only 23.4%, 6.2%, and 24.4% of
farmers in the state used forward contracting, hedging, and crop insurance, respectively,
as risk reduction strategies, suggesting these tools may be ineffective. Farmers described
diversification across crops as an easily implemented and effective strategy for managing
revenue risk. This view was supported in an earlier study (Blank 1990), where the author
showed there was an optimal amount of crop diversification among crop portfolios, and that
this risk management strategy was always preferable to specialization.
3 Mathematical models applied in agriculture
Increasing environmental problems in agriculture urge policy-makers to develop instruments
to reduce and control the pollution caused by current intensive farming practices. These
measures should be both effective from the environmental point of view, which is a public
goal, as well as acceptable at the farm level with regard to private goals such as income and
continuity at the farm. Operations research is an effective tool for modeling the complex
interaction of production intensity, environmental aspects and farm income.Agriculture is one of the fields where mathematical models of operations research were
first used and also where they have been most widely applied. The number of mathematical
models in agriculture has rapidly grown in the last decades, due to the impressive devel-
opment of personal computers and commercial software programs. One of the first papers
that apply linear programming to agricultural decision making situations is Heady (1954).
The early models were formulated at the farm level. One of the first books dedicated to the
application of linear programming to agriculture was Beneke and Winterboer (1973). For
historical notes regarding the applications of operations research models to agriculture and
other applications we recommend (Weintraub et al. 2001, 2007). Simulation models havealso been used in the last decades at the farm level to assess the economic impact of several
agricultural policies. Until recently, most of the farmers have made intensive use of chemical
inputs in order to obtain higher yields at low costs. Unfortunately this policy had produced
many negative side effects (or, as economists put it, negative externalities) for the environ-
ment. At present there exists a trend to develop mathematical models that are able to evaluate
the economic impact of environment damages and thus to contribute to the development of
a sustainable agriculture.
The great majority of the problems connected to sustainable agriculture have a multi-
criteria character. The book (Romero and Rehman 2003) is a comprehensive monograph for
the state of the art of multi-criteria analysis in agriculture until 2003. The concept of sustain-able agriculture supposes harmonization or simultaneous realization of the objectives con-
nected to economic growth, environment and social development. The mathematical models
that take into account both objectives mentioned above are scarce. We mention a few of them
below. In Wossink et al. (1992) an extension of the linear programming optimization model
is employed in farm economics with an environmental component to analyze and evaluate
the effects of alternative environmental policy instruments for agriculture. The application
presented concerns the potential role of technical innovations and of input levies to reduce
biocide use in crop production. In Johnson et al. (1991) a crop simulator model, called
CERES, was linked to a dynamic optimization model in order to determine the optimum
application of water and fertilizers that maximize the gross margin of yield. In Zekri and
Herruzo (1994) the authors use a combination between a crop simulator model and a mixed
multiobjective programming model in order to assess the effects of an increase of nitrogen
prices and drainage water reduction. In Teague et al. (1995) the authors use the EPIC-PST
model to predict the environmental risks from the use of pesticide and nitrates. A combina-
tion of these results with a Target MOTAD optimization model are used for the assessment
of the trade-offs that exist between income and an index that linked both risks associated to
the use of pesticides and nitrates. In Annetts and Audsley (2002) a multiple objective linear
programming model that allows maximization of the return and environmental outcomes is
presented.
A method that has been successfully applied to crop planning is the Interactive MultipleGoal Linear Programming (IMGLP). In Nidumolu et al. (2007) an IMGLP model was devel-
oped that considers objectives of multiple stakeholders, i.e. different farmer groups, district
agricultural officers and agricultural scientists for agricultural land use analysis. The anal-
ysis focuses on crop selection; considering irrigated and non-irrigated crops such as rice,
sugarcane, sorghum, cotton, millet, pulses and groundnut. Interests of the most important
stakeholders, farmers, policy makers and water users association are investigated. Important
objectives of the farmers are increased income and retaining paddy area; of the policy mak-
ers (Agricultural Department) increased farmers’ income, maintaining rural employment,
improve water-use efficiency, reduce fertilizer and biocide use and discourage farmers from
cultivating marginal lands; of the water users association optimizing water use. Scenarioshave been constructed by combining objectives and constraints. In C.T. Hoanh et al. ( 2000)
a multiple goal linear programming (MGLP) model was included in a land use planning and
a DSS system (LUPAS). It was used to investigate the consequences for land use of vari-
ous decisions taken under the economic reform. The incorporation of methods and tools for
multi-decision-level (field, farm, and region) analysis made LUPAS a powerful and versatile
DSS for the integrated resource management and policy design.
In general these models do not take into account simultaneously uncertainty and en-
vironmental issues. In general, decision processes in agriculture are complicated by the
uncertainty surrounding important components such as weather, product prices, and gov-
ernment policies. Biological and economic management processes are basic to the success
of agricultural enterprises. Many applications of risk analysis in agriculture show the im-
pact of uncertainty on agricultural decision making. The uncertainty from the agriculture
problems is modeled with the help of probability theory. Many of the practical problems
that occur in agriculture are stochastic programming problems with multiple objectives. In
practice, in the process of mathematical modeling, one cannot take into account all the fac-
tors 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.
Among the mathematical models applied in agriculture one can quote prediction models,
crop planning models, optimal selection of fertilizer/pesticide models, crop rotation mod-els etc. An important mathematical instrument which was successfully applied to model-
ing the problems from agriculture was portfolio theory. The above mentioned theory was
developed as a result of the research in the domain of financial management. Its aim is
the elaboration of a quantitative analysis of how investors can diversify their portfolio in
order to minimize risk and maximize returns. The theory was introduced in 1952 by Uni-
versity of Chicago economics student Harry Markowitz, who published his doctoral the-
sis, “Portfolio Selection” in the Journal of Finance (Markowitz 1952). The application of
portfolio theory for finding an optimal allocation of agricultural land is popular in the liter-
ature. Since the 1950s, agricultural economists have adapted many popular financial port-
folio selection rules to the farm manager’s land allocation problem. The first application
of the portfolio theory to crop planning goes back to Freund (1956). In his approach Fre-
und defined the risk of an agricultural enterprise through the variability of its returns, mea-
sured by variance. This implies the use of quadratic programming for finding the optimal
crop patterns. In Hazell (1971) the author had changed the risk measure for the farm re-
turn. The variance was replaced by the mean absolute deviation. As a result, Hazell showed
that the crop planning problem is equivalent to a linear programming problem. Single in-
dex portfolio models were used to farm planning in the papers (Collins and Barry 1986;
Turvey et al. 1988). In Newbery and Stiglitz (1981), Schaefer (1992), Hardaker et al. (2004),
Hazell and Norton (1986), Blank (2001) were presented or applied various variants of port-
folio theory to the land allocation decisions. In Collender (1989), Reeves and Lilieholm(1993), Romero (1976, 2000) were studied several models for resources allocation in agri-
culture that are taking into account specific risks. Mathematical models that take into ac-
count farmers decisions and climate change were studied in Lewandrowski and Brazee
(1993). In Rafsnider et al. (1993) portfolio analysis, using nonlinear risk programming, was
applied to identify the risk-efficient crop portfolios for a group of subsistence farmers. In
Fafchamps (1992), a simple theoretical model of crop portfolio choice when the revenue of
individual crops is correlated with consumption prices is studied. In Roche and McQuinn
(2004) the authors investigate the efficient allocation of land in a mean-variance sense. They
use portfolio theory in order to examine the potential implications on the land allocation
decision of the 2002 EU Commission’s proposed mid-term reform of the Common Agricul-tural Policy related to decoupled payments.
Applications of portfolio theory to biodiversity conservation were studied in Figge
(2004). For other references regarding applications of portfolio theory to agriculture see
Radulescu et al. (2006, 2010).
4 A crop planning model for sustainable agriculture
In this section a mixed integer programming model with multiple objectives for crop plan-
ning in agriculture is formulated. The model takes into account weather risks, market risks
and environmental risks. Input data include historical land productivity data for various
crops and soil types, yield response to fertilizer/pesticide application, unitary costs of fertil-
izers/pesticides, costs for crops cultivation (without using fertilizers/pesticides) on the plots.
The application of fertilizers/pesticides to agricultural crops is desirable since they con-
tribute to the growth of agricultural production. On the other side the application of fertiliz-
ers/pesticides in great quantities, over some levels bring damages to environment and human
health. In order to protect the environment and of course the people’s health we shall take
into account two kinds of levels: desirable levels and maximum admissible levels for the
application of chemical inputs (fertilizers/pesticides). If the quantity of the chemical input isunder the desirable level then no penalty is paid by the farmer. If the quantity of the chemi-
cal input lies between the desirable level and the maximum admissible level then monetary
penalties proportional to the amount that it overcomes the desirable level must be paid by
the farmer.
Consider that an agricultural region is divided into several agricultural subregions.
Suppose that:
• the soil quality of a subregion is homogeneous
• the climate in a subregion is the same
• historical data on land productivities in each agricultural subregions is available for agiven set of crops
For example a subregion may have non-irrigated good land quality, other region may have
irrigated medium land quality, etc.
Consider a farm located in an agricultural region as described above, which has an agri-
cultural land divided into several plots. Let P 1, P 2, . . . , P m be the plots from the farm’s land.
Suppose that each plot P j belongs to an agricultural subregion and consequently the soil
quality of each plot is homogeneous. We consider that if a plot is cultivated then it is culti-
vated with the same crop. Denote by S j the area of the plot P j , j = 1, 2, . . . , m. We consider
that the farmer has to choose a crop plan from n crops C1, C2, . . . , Cn, that is to make an al-
location of crops to plots. In order to obtain high yields the farmer uses fertilizers/pesticides.Denote by F 1, F 2, . . . , F k the set of the k fertilizers/pesticides used by the farmer. For the
fertilizer/pesticide F r denote by q1r the desirable level for the application of F r . A crop
plan that uses a quantity of fertilizer/pesticide under the desirable level is not subjected to
monetary penalizations. An overcome of the desirable level is subjected to the application
of environmental taxes. These taxes and the desirable and the maximum admissible levels
may be established at various levels of decision making: local level, regional level, country
level etc. Their amount depends on the existing level of pollution and on the community
aspiration for maintaining a clean environment.
In our paper will shall consider appropriate monetary penalties for the application of
fertilizers/pesticides that are proportional to the amount that exceeds the desirable level. Weshall denote by q2ir the maximum admissible level for the application of F r per one hectare
cultivated with crop Ci . The introduction of the pollution levels in the management of risk
environment goes back to Qiu et al. (2001). For mathematical models that use the pollution
levels in production planning see Radulescu et al. (2009).
Let U i be the Cartesian product of intervals [0, q2ir ], that is:
U i = [0, q2i1] × [0, q2i2] × · · · × [0, q2ik ].
Consider the probability space ( , K , P ). Denote I = {1, 2, . . . , n}, J = {1, 2, . . . , m},
K = {1, 2, . . . , k}. For every i ∈ I, j ∈ J we define the plot productivity functions cij : ×U i → R+ and the market price functions bi : → R+. Thus if q is a vector from U i then the
function φq(ω) = cij (ω, q), ω ∈ is a random variable. Analogous all the functions bi are
random variables. cij (·, q) represents the quantity (in kg) of crop Ci that can be produced
on one hectare of plot P j and bi the market price of one kg of crop Ci .
In this paper the effect of crop rotation is neglected. That is if one cultivates the same crop
repeatedly on the same plot over the time, the yield decrease. The effect of crop rotation is
taken into account in the papers (Alfandari et al. 2009; Castellazzi et al. 2008; Detlefsen
and Jensen 2007; Dogliotti et al. 2003; El-Nazer and McCarl 1986; Haneveld and Stegeman
2005; Mimouni et al. 2000; Santos et al. 2008; Vizvári et al. 2009).
One can easily see that in our model the plot productivity functions do not depend on the
crops cultivated previously on that plot.
Let aij be the sum of money used by the farmer in order to cultivate one hectare of plot
P j with crop Ci without using fertilizers/pesticides. For every i ∈ I , j ∈ J , r ∈ K denote
by:
• d r —the cost of one kg of fertilizer/pesticide F r .
• wr —the penalization coefficient for the case the quantity of fertilizer/pesticide F r over-
comes the desirable level q1r .
• yijr —the decision variable representing the quantity of fertilizer/pesticide F r applied to
one hectare of plot P j cultivated with crop Ci . Denote yij = (yij1, yij2, . . . , yijk ).
• xij —the decision variable that takes the value 1 if the crop Ci is cultivated on plot P j and
takes value zero if the crop Ci is not cultivated on the plot P j .
• [M 1, M 2] the range for the sum of money available for investment.
• Qi the inferior bound for the expected yield of crop Ci necessary to be obtained. It may
Mathematical programming problems of the above type are very difficult to solve since
they exhibit exponential complexity resulting from the presence of integer variables. Tradi-tional approaches that apply in pure integer programming are not very helpful since the ex-
istence of continuous variables complicates their solution. Recently, simulation-based meth-
ods and heuristic algorithms have been successfully used for solving such problems. In the
literature there is a small number of papers dealing with mixed integer quadratic program-
ming problems (Lazimy 1982; Tziligakis 1999). In Lazimy (1982) is described an approach
for solving such programs based on the generalized Benders’ decomposition. A new equiv-
alent formulation that renders the program tractable is developed, under which the dual
objective function is linear in the integer variables and the dual constraint set is indepen-
dent of these variables. In Tziligakis (1999) an approach based on the relaxation method is
developed for solving mixed integer quadratic programs. A heuristic algorithm is built for
providing tight lower and upper bounds for the mixed integer problem.
Starting from the above multiobjective programming model for the crop planning we
formulate several single objective problems. More precisely we shall consider a minimum
environmental risk problem, a maximum return problem and a minimum return risk prob-
The models mentioned above are very useful in practice since they help farmers to make
optimal decisions in complex situations in which weather and economic risks as well as
environmental constraints may be involved.
The application in practice of the models supposes that there are records (historical data)
on the land productivities for various crops, plots and the records on the impact of fer-tilizers/pesticides on the crops. One of the key assumptions is that a sufficient amount of
(historical) data is available. In reality, it is unlikely that this assumption holds. In order to
solve this limited data problem one can use bootstrapping methods that increase the number
of data through simulation. Another problem is the problem of establishing the environmen-
tal levels. In an agricultural region the values of the environmental levels are supposed to be
fixed by the region or by the state environmental protection department. It strongly depends
on the degree of pollution in the agricultural region. The value of this level may decrease in
a polluted area and increase in a low polluted area.
5 The minimum environmental risk problem
In the framework of this problem the farmer tries to find an optimal allocation of crops to
plots and an optimal plan for fertilizer and pesticide application that minimize the environ-
mental risk taking into account that:
• the financial risk, that is Var(f 5(x, y, ·)), is smaller than a prescribed level τ • the expected return is greater than a given level W
• the cost of cultivation and application of the fertilizers and pesticides lies in the range
• the matrix Q2 = (q2ir ) of maximum admissible levels for fertilizer/pesticide application
• the vector w = (w1, w2, . . . , wk ) of monetary penalties for fertilizer/pesticide application
• the vector q1 = (q11, q12, . . . , q1k ) of desirable levels for fertilizer/pesticide application
• the vector Q = (Q1, Q2, . . . , Qn) of demand levels for crops
• the vector S = (S 1, S 2, . . . , S m) of areas of plots• the vector d = (d 1, d 2, . . . , d k ) of fertilizer/pesticide prices
The user parameters are:
• the budget limits M 1 and M 2• the lower limit for the expected return (the number W )
• the upper limit for the financial risk (parameter τ )
The decision variables are the allocation matrix x = (xij ) whose entries are 0 and 1 and the
matrix y = (yij ) where yij = (yij1, yij2, . . . , yijk ) ∈ U i , i ∈ I , j ∈ J .
The objective map of this problem is the environmental risk, that is the overall monetary
penalizations paid by the farmer for his crop plan. It is equal to
f 3(x, y) =
kr=1
wr
n
i=1
mj =1
yijr S j − q1r
+
.
The number of variables in the above problem is equal to mnk + mn + k and the number of
constraints is equal to 2mnk + m + n + 2k + 3. The number of binary variables is equal to
mn.
We consider 3 crops and two fertilizers urea and triple superphosphate. Crop C1 is wheat,
crop C2 is corn, crop C3 is barley. The demand levels (components of vector Q) are: Q1 =17000 kg wheat, Q2 = 7000 kg corn, Q3 = 10000 kg barley. We consider 5 plots. The areas
of the plots are S 1 = 9 ha, S 2 = 15 ha, S 3 = 12 ha, S 4 = 11 ha, S 5 = 10 ha. The period of
time considered is 1995–2006.
By solving two mixed linear integer problems (see the preceding paragraph) we com-
pute M max2 and M min
1 . We find M max2 = 44131 euros and M min
1 = 18369 euros. We choose
M 1 = M min1 and M 2 = M max
2 . By solving two other mixed integer problems (see the pre-
ceding paragraph) we compute τ min and τ max. For 5 values of τ that is for τ 1 = 13824600,
τ 2 = 23982700, τ 3 = 34140800, τ 4 = 44298900, τ 5 = 54457000 are computed W min(τ i ) and
W max(τ i ), i = 1, 2, . . . , 5. For 6 values of W chosen in the intervals [W min(τ i ), W max(τ i )] is
solved the minimum environmental risk problem. The results are displayed in Table 1. Inthe fourth column of Table 1 are displayed Varcal, the values of the risk computed for the
optimal crop plan obtained with the help of the parameters τ and W from the columns 2 and
3. Analogously in the fifth and the sixth column are displayed Wcal, the values of the return
computed for the optimal crop plan obtained with the help of the parameters τ and W , and
Mcal, the sum invested in the optimal crop plan. In the last column of Table 1 are displayed
the optimal values of the minimum environmental risk problem that is the environmental
penalties paid for the crop plan.
For τ = 54457000 the optimal allocations for the various values of W are displayed in
Table 2. The optimal values of the objective function are displayed in the seventh column.
In the eighth column are displayed the expenses made with the fertilizers computed for the
optimal crop plan. In the last column are displayed the rates of return (the ratio between
the return and the cultivation costs) computed for the optimal crop plan. The cells that are
empty in the table mean that the plots are not cultivated (fallow).
From Table 2 one can note that for the first three values of W , the values of the objective
function are zero since the no fertilizer is used. An analysis of the monetary penalizations
the second column. The cells that are empty in the table mean that the plots are not cultivated
(fallow). In the second row are displayed the optimal values of the objective function, that is
the penalties paid for the use of fertilizers and pesticides. In the third row are displayed the
expenses made with fertilizers and pesticides. From Table 3 one can note that when the rate
of return increases, the number of fallows decreases and also the area cultivated with sun-flower increase. An explanation for this thing is the fact that the market price for sunflower
is higher than the other prices, whence the cultivation of sunflower is more profitable.
10 Conclusions
Agriculture is a risky business and a source of pollution for the environment. In this paper we
have presented a multiobjective programming model that tries to solve the tradeoff between
the economic and environmental objectives of the agricultural production. The aim of the
model is to find an optimal allocation of crops to plots and an optimal application rate of the chemicals such that the financial risk and monetary penalties paid for the environmental
pollution are minimized and expected return of the agricultural production is maximized.
Starting from the multiobjective model several single objective models are formulated:
the minimum environmental risk model, the maximum expected return model and the min-
imum financial risk model. By introducing some supplementary variables we have shown
that the above models are equivalent to mixed integer quadratic models. For the minimum
environmental risk model two numerical examples are analyzed.
Acknowledgements We especially thank to four anonymous referees for helpful comments and critical
suggestions.
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