a portfolio theory approach

8/6/2019 A Portfolio Theory Approach

http://slidepdf.com/reader/full/a-portfolio-theory-approach 1/22

Ann Oper Res

DOI 10.1007/s10479-011-0902-7

A portfolio theory approach to crop planning

under environmental constraints

Marius Radulescu · Constanta Zoie Radulescu ·

Gheorghita Zbaganu

© Springer Science+Business Media, LLC 2011

Abstract This paper presents a multiobjective model for crop planning in agriculture. The

approach is based on portfolio theory. The model takes into account weather risks, market

risks and environmental risks. Input data include historical land productivity data for various

crops, soil types and yield response to fertilizer/pesticide application. Several environmental

levels for the application of fertilizers/pesticides, and the monetary penalties for overcoming

these levels, are also considered. Starting from the multiobjective model we formulate sev-

eral 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. Two numerical results for the minimum environmental risk prob-

lem are presented.

Keywords Crop planning · Portfolio theory · Multiobjective model · Environmental risk ·

Environmental level · Mixed integer model

M. Radulescu ()

Institute of Mathematical Statistics and Applied Mathematics, Casa Academiei Romane,

Calea 13, Septembrie nr. 13, 050711 Bucharest 5, Romania

e-mail: [email protected]

C.Z. Radulescu

National Institute for Research and Development in Informatics, 8-10 Averescu Avenue,

011455 Bucharest 1, Romania


C.Z. Radulescu


G. Zbaganu

Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14,

010014 Bucharest, Romania


mailto:[email protected]










Ann Oper Res

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,



Ann Oper Res

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



Ann Oper Res

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



Ann Oper Res

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.



Ann Oper Res

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

be thought as the expected demand for crop Ci .



Ann Oper Res

In our model we shall make the following assumption:

cij (ω, q) = cij0(ω) +

k

r=1

cijr (ω)qr where q = (q1, q2, . . . , qk) ∈ U i and ω ∈ .

The meaning of the coefficients cijr is the following:

• cijr —the additional quantity (in kg) of crop Ci obtained from one hectare of plot P j when

one kg of fertilizer/pesticide F r is used.

The above assumption is a simplification since, in reality, the impact of an additional quan-

tity of fertilizer F r does depend on the levels of other fertilizers F r . In some cases this

assumption could be restrictive as there can be synergy of fertilizers.

• cij0—the quantity of crop Ci that it is obtained from one hectare of plot P j when no

fertilizer/pesticide are used.

We shall call a crop plan a couple (x, y) where x = (xij ) is a n × m matrix with entries 0–1

and y = (yij ) where yij = (yij1, yij2, . . . , yijk ) ∈ U i , i ∈ I , j ∈ J .

The environmental constraints in the model are represented by the inequalities:

0 ≤ yijr ≤ q2ir xij , i ∈ I, j ∈ J, r ∈ K. (1)

This means that the quantity of fertilizers/pesticides is smaller than an upper limit q2ir if the

plot P j is cultivated with crop Ci . The quantity of fertilizers/pesticides is equal to zero if the

plot P j is not cultivated with crop Ci . When some crop does not need the fertilizer/pesticide

F r one can take q2ir = 0. From the above inequality one can easily see that

yijr = xij yijr , i ∈ I, j ∈ J, r ∈ K. (2)

Indeed, if xij = 0 then from inequality (1) it follows that yijr = 0 hence equality (2) holds.

In the case xij = 1 the equality (2) obviously holds.

The yield of crop Ci obtained from the plot P j when the decisions are given by the crop

plan (x, y) is equal to

xij S j

cij0 +

kr=1

cijr yijr

= S j

cij0xij +

kr=1

cijr yijr

.

Take yij0 = xij for every i ∈ I , j ∈ J . The yield of crop Ci when the decisions are given by

the crop plan (x, y) is equal to

m

j =1

k

r=0

cijr yijr S j .

The sales revenue when the decisions are given by the crop plan (x, y) is equal to the random

variable

f 1(x, y, ·) =

ni=1

mj =1

kr=0

bi cijr S j yijr .



Ann Oper Res

The demand constraints are represented by the inequalities:

m

j =1

k

r=0

E(cijr )yijr S j ≥ Qi , i ∈ I.

The budget constraints are represented by the inequalities: M 1 ≤ f 2(x, y) ≤ M 2.

The restrictionsn

i=1 xij ≤ 1, j ∈ J show that every plot is cultivated with at most one

crop.

The farmer intends to obtain optimal production plans that minimize the environmen-

tal penalties, minimize the financial risk and maximize the expected return. According to

the above requirements the multiobjective mixed integer programming model for the crop

planning is the following:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

min k

r=1

wr

ni=1

mj =1

yijr S j − q1r

+

,

min

n

i=1

mj =1

kr=0

nα=1

mβ=1

kγ =0

ρijrαβγ yijr yαβγ S j S β

,

max

n

i=1

mj =1

(E(bi cij0) − aij )yij0 +

kr=1

(E(bi cijr ) − d r )yijr

S j

,

M 1 ≤ f 2(x, y) ≤ M 2,

mj =1

kr=0

E(cijr )yijr S j ≥ Qi , i ∈ I,

0 ≤ yijr ≤ q2ir yij0, i ∈ I, j ∈ J, r ∈ K,n

i=1

yij0 ≤ 1, for every j ∈ J,

yij0 ∈ {0, 1}, i ∈ I, j ∈ J.

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-

lem.



Ann Oper Res

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

[M 1, M 2]

• the demand constraints are satisfied

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

min

k

r=1

wr

n

i=1

mj =1

yijr S j − q1r

+

,

Var(f 5(x, y, ·)) ≤ τ,

E(f 4(x, y, ·)) −k

r=1

wr

n

i=1

mj =1

yijr S j − q1r

+

≥ W,

M 1 ≤ f 2(x,y) ≤ M 2,m

j =1

kr=0



i=1


yij0 ∈ {0, 1}, i ∈ I, j ∈ J.

With an introduction of some additional variables the objective function can be transformed

in a linear map. Thus the minimum environmental risk problem is equivalent to the following

mixed integer quadratic problem with linear objective function and linear and quadratic



Ann Oper Res

constraints: ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

min

k

r=1

wr zr

,

ni=1

mj =1

yijr S j − q1r ≤ zr , r ∈ K,

Var(f 5(x, y, ·)) ≤ τ,

E(f 4(x, y, ·)) −

kr=1

wr zr ≥ W,

M 1 ≤ f 2(x, y) ≤ M 2,m

j =1

k

r=0


0 ≤ yijr ≤ q2ir yij0, 0 ≤ zr , i ∈ I, j ∈ J, r ∈ K,n

i=1


yij0 ∈ {0, 1}, i ∈ I, j ∈ J.

Recall that the land productivity coefficients cijr are random variables. They incorporate

the weather risks. Historical data on the response rate to fertilizer/pesticide application, that

is the numbers cijrt can be considered as realizations of the random variables cijr . The

coefficients bi whose meaning is market price of crops are random variables that incorporate

the market risk. Historical data on market price of crops, that is the numbers bit = the market

price for one unit of crop Ci at moment t can be considered as realizations of the random

variables bi .

Taking into account the above things, we shall approximate the variance of the return

Var(f 5(x, y, ·)) =

ni=1

mj =1

kr=0

nα=1

mβ=1

kγ =0


with the sample variance, that is with the number

R(x, y) =

ni=1

mj =1

kr=0

nα=1

mβ=1

kγ =0


where we denoted:

ρijrαβγ =1

T

T

t =1

bit cijrt bαt cαβrt −1

T 2T

t =1

bit cijrt T

t =1

bαt cαβrt .

We shall approximate the means E(bi cijr ), E(cijr ) by their sample means. Here T is the

number of moments of time in the time horizon considered.

We shall approximate the minimum environmental risk problem with a similar problem

in which the variance of the return is replaced by its sample variance and the means by their

sample means.



Ann Oper Res

6 The maximum return problem

In the framework of the maximum return problem the farmer tries to find crop plan (x, y)

that maximize the expected net return taking into account that

• the financial risk, that is Var(f 5(x, y, ·)), is smaller than a prescribed level τ

• the cultivation cost and application of the fertilizers and pesticides lies in the range

[M 1, M 2]

• the demand constraints are satisfied⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

max

E(f 4(x, y, ·)) −

kr=1

wr

n

i=1

mj =1

yijr S j − q1r

+

,

Var(f 5(x, y, ·)) ≤ τ,

M 1

≤ f 2

(x,y) ≤ M 2

,m

j =1

kr=0



i=1


yij0 ∈ {0, 1}, i ∈ I, j ∈ J.

7 The minimum financial risk problem

In the framework of this problem the farmer tries to find an optimal crop plan (x, y) that

minimizes the financial risk taking into account that

• the expected net return is greater than a given level W

• the cultivation cost and application of the fertilizers and pesticides lies in the range

[M 1, M 2]

• the demand constraints are satisfied⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

min(Var(f 5(x, y, ·))),

E(f 4(x, y, ·)) −

kr=1

wr

n

i=1

mj =1

yijr S j − q1r

+

≥ W,

M 1 ≤ f 2(x, y) ≤ M 2,m

j =1

kr=0


0 ≤ yijr ≤ q2ir yij0, i ∈ I, j ∈ J, r ∈ K,

ni=1


yij0 ∈ {0, 1}, i ∈ I, j ∈ J.

With an introduction of some additional variables the minimum financial risk problem

can be transformed into a mixed integer problem with a quadratic objective function and



Ann Oper Res

linear constraints. ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

min(Var(f 5(x, y, ·))),

E(f 4(x, y, ·)) −

k

r=1

wr zr ≥ W,

ni=1

mj =1

yijr S j − q1r ≤ zr , r ∈ K,

M 1 ≤ f 2(x, y) ≤ M 2,m

j =1

kr=0


0 ≤ yijr ≤ q2ir yij0, 0 ≤ zr , i ∈ I, j ∈ J, r ∈ K,n

i=1


yij0 ∈ {0, 1}, i ∈ I, j ∈ J.

8 The decision process for the minimum environmental risk problem

An important problem in the decision-making process of the choice of a crop plan is the

existence of some indicators that will guide the farmer in the selection of the parameters of

the problem: M 1, M 2, Qi , i = 1, 2, . . . , n, τ and W . These indicators can be computed by

solving several mixed integer problems. Denote by X1 the set of all y = (yijr )i∈I,j ∈J,r ∈K∪{0}

such that its entries satisfy the last three constraints of the minimum environmental risk

problem. Consider the problem

(P 1) max{f 2(y) : y ∈ X1}.

Denote by M max2 its optimal value. This value represents an upper bound for the sum of

money the farmer can invest in some crop plan. The farmer may choose the parameter M 2smaller or equal than M max

2 . Denote by M max1 the optimal value of the problem

(P 2) max{f 2(y) : y ∈ X1, f 2(y) ≤ M 2}.

For every i ∈ I consider the problem

(P 3(i)) max

m

j =1

kr=0

E(cijr )yijr S j : y ∈ X1

and denote by Qmaxi its optimal value. The farmer can choose the parameters Qi ,

i = 1, 2, . . . , n such that Qi ≤ Qmaxi , i = 1, 2, . . . , n. Denote

X2 =

y ∈ X1 :

mj =1

kr=0

E(cijr )yijr S j ≥ Qi for every i ∈ I

.

Of course the farmer has to choose the parameters Qi , i = 1, 2, . . . , n such that the set X2 is

non-empty. This choice may be done as a result of a sequence of trial error experiments.



Ann Oper Res

Consider the problem

(P 4) min{f 2(y) : y ∈ X2}.

Denote by M min

1

its optimal value. Then the farmer has to choose the parameter M 1 in the

interval [M min1 , M max

1 ]. Suppose that the farmer has chosen the parameters M 1, M 2 and Qi ,

i = 1, 2, . . . , n and wants to determine a range in which he must choose the parameter τ .

Consider the problems:

(P 5) min{Var(f 5(y)) : y ∈ X2},

(P 6) max{Var(f 5(y)) : y ∈ X2}.

Denote by τ min (respectively by τ max) the optimal values of the problems (P 5) (respectively

(P 6)). Then the farmer has to choose the parameter τ in the interval [τ min, τ max]. Suppose

now that the farmer has already chosen the parameter τ in the above mentioned interval.Consider the problems:

(P 7(τ)) min

E(f 4(x, y, ·)) −

kr=1

wr

n

i=1

mj =1

yijr S j − q1r

+

: y ∈ X2, Var(f 5(y)) ≤ τ

,

(P 8(τ)) max

E(f 4(x, y, ·)) −

kr=1

wr

n

i=1

mj =1

yijr S j − q1r

+

: y ∈ X2, Var(f 5(y)) ≤ τ

and denote by W min(τ ) and W max(τ ) their optimal values. The farmer has to choose the

parameter W in the interval [W min(τ),W max(τ )]. Suppose that the farmer has already chosen

the parameter W in the above mentioned interval. Consider the problem

min

k

r=1

wr

n

i=1

mj =1

yijr S j − q1r

+

: y ∈ X2, Var(f 5(y)) ≤ τ,

E(f 4(x, y, ·)) −

k

r=1

wrn

i=1

m

j =1

yijr S j − q1r+

≥ W .

This problem is the minimum environmental risk problem. We have thus described a pro-

cedure for the selection of the parameters of the problem. This procedure allows the farmer

to select parameters of the model in such a way that the solutions of the model are feasible.

Consequently this procedure brings an argument to the model applicability and validity.

9 Numerical example for the minimum environmental risk problem

In the following we shall analyze a numerical example for the minimum environmental risk problem. The input data are represented by:

• the matrix A = (aij ) of cultivation costs

• the matrix B = (bit ) of historical market prices for crops

• the four-dimensional matrix C = (cijrt ) of crop yields response rate to fertilizer/pesticide

application



Ann Oper Res

• 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



Ann Oper Res

Table 1 The optimal values of the environment risk problem for various values of the parameters τ and W

Nr. crt. τ W Varcal Wcal Mcal Env. risk

1 13824600 −1066.80 11949412 −1066.80 30127.50 0.00

2 13824600 −251.49 10806292 −251.49 24440.39 8.773 13824600 563.81 13474417 563.81 24168.77 33.29

4 13824600 1379.12 12071211 1379.12 22485.46 71.52

5 13824600 2194.43 13264525 2194.43 22699.51 122.49

6 13824600 3009.74 13824600 3009.74 22000.33 196.06

7 23982700 −1199.59 22924320 −1199.59 37707.50 0.00

8 23982700 −272.23 9308453 −272.23 21917.98 2.07

9 23982700 655.13 10833172 655.13 22329.94 34.50

10 23982700 1582.49 12404341 1582.49 22533.28 82.91

11 23982700 2509.84 16475684 2509

.84 24063

.14 141

.83

12 23982700 3437.20 23301521 3437.20 21409.16 171.75

13 34140800 −1199.59 22924320 −1199.59 37707.50 0.00

14 34140800 −110.28 9590329 −110.28 21994.67 6.10

15 34140800 979.02 11375299 979.02 22399.51 51.06

16 34140800 2068.32 15911657 2068.32 23940.48 112.62

17 34140800 3157.63 31298718 3157.63 30562.25 175.53

18 34140800 4246.93 33297769 4246.93 38510.86 379.58

19 44298900 −1199.59 22924320 −1199.59 37707.50 0.00

20 44298900 −44.69 8879292 −44.69 20926.40 7.52

21 44298900 1110.22 11600301 1110.22 22427.70 57.77

22 44298900 2265.12 16160842 2265.12 23995.15 125.64

23 44298900 3420.02 32113475 3420.02 30873.77 206.11

24 44298900 4574.93 44298900 4574.93 39752.16 380.71

25 54457000 −1199.59 22924320 −1199.59 37707.50 0.00

26 54457000 18.12 8986053 18.12 20956.87 9.13

27 54457000 1235.83 11818648 1235.83 22454.68 64.19

28 54457000 2453.54 28973060 2453.54 30313.50 136.84

29 54457000 3671.25 32573501 3671.25 31000.94 236.38

30 54457000 4888.96 54212999 4888.96 40517.36 392.71

for environmental pollution (seventh column) shows that: the greater is the return desired

by the farmer, the greater is the amount of fertilizer used, therefore the greater are the mon-

etary penalties paid for the environment pollution. However the rate of return is in general

increasing with respect to the parameter W .

The efficient frontier of the minimum environmental risk problem is displayed in Fig. 1.

The efficient frontier is the graph of the function that associates to values of W the optimal

value of the objective function. It shows how the variation of the target profit influences the

amount of the monetary penalties paid for overcoming the environmental levels.

The graph above is the graph of the fertilizer expenses versus the parameter W .

The numbers M max2 , M max

1 , M min1 and Qmax

i , i ∈ I that define the ranges of parameters

M 1, M 2 and Qi , i ∈ I were computed with ILOG-CPLEX package for MIP. The numbers

τ min, τ max, W min(τ ) and W max(τ ) that define the ranges of parameters τ and W were com-



Ann Oper Res

Table 2 Optimal allocations of crops to plots and optimal values of the objective function for various values

of parameter W

W Plot1 Plot2 Plot3 Plot4 Plot5 Env. Risk Exp. Fert. Rate return

−1199.59 corn wheat corn corn barley 0 0 −0.03

−895.16 wheat corn barley 0 0 −0.02

−590.73 wheat corn barley 0 0 −0.02

−286.31 barley wheat corn 1.73 49.1 −0.01

18.12 barley wheat corn 9.10 189.05 0.00







2149.11 corn wheat corn 117.97 825.42 0.09

2453.54 barley corn wheat corn 136.84 919.50 0.08





3975.68 barley corn corn wheat corn 264.08 1785.18 0.10




Fig. 1 The efficient frontier for the minimum environmental risk problem

puted with DICOPT2X-C for MINLP. The optimization problems were solved on a com-

puter with 2 GHz Intel Core 2 Duo processor and Windows XP operating system. Numerical

examples for various dimensions were performed (n varied from 5 to 7 crops, m varied from



Ann Oper Res

Table 3 Optimal allocations of crops to plots for 5 values of parameter W

Area W 1 = 6520 W 2 = 8540 W 3 = 23601 W 4 = 38662 W 5 = 53723

Fob 0 102 569 1137 2073

Exp. fert. 0 1970 4908 9200 13512

plot1 9 rapeseed rapeseed barley barley barley

plot2 15 corn sunflower sunflower sunflower sunflower

plot3 12 barley barley barley corn corn

plot4 11 sunflower sunflower

plot5 10 wheat sunflower sunflower sunflower sunflower

plot6 13 corn corn corn sunflower

plot7 12 corn corn sunflower sunflower

plot8 15 barley barley barley sunflower sunflower

plot9 10 corn corn sunflowerplot10 15 corn sunflower sunflower sunflower


plot12 9 rapeseed rapeseed barley


plot14 12 wheat sunflower sunflower

plot15 11 corn corn sunflower

plot16 10 sunflower sunflower sunflower sunflower

plot17 13 corn barley sunflower sunflower sunflower

plot18 12 wheat sunflower sunflower corn corn

plot19 15 sunflower sunflower sunflower

plot20 10 corn corn corn corn

plot21 15 wheat wheat sunflower sunflower sunflower

plot22 10 corn wheat wheat sunflower

plot23 9 wheat corn sunflower sunflower sunflower

plot24 15 corn corn corn sunflower sunflower

plot25 12 corn corn corn corn


plot27 10 corn wheat wheat sunflower

plot28 13 sunflower sunflower sunflower sunflower sunflower

plot29 12 corn corn corn sunflower sunflower

plot30 15 corn sunflower sunflower wheat

10 to 120 plots and k varied from 2 to 4 fertilizers/pesticides). The CPU time for obtaining

solutions of optimization problems varied from one minute to 35 minutes. Some of the so-

lutions obtained were suboptimal. The numerical experiments performed for the minimum

environmental risk problem showed that GAMS solvers are suitable for the resolution of

realistic instances of the problems presented in the paper.

We consider in the following another numerical example with 5 crops: corn, wheat,

barley, sunflower and rapeseed, 30 plots, two fertilizers (urea and triple superphosphate)

and two herbicides (Banvel 480 S and Dual Gold 960 EC). We take M 1 = 10000 euro and

M 2 = 70000 euro. For τ = 859234771300 the optimal allocations of crops to plots for var-

ious values of W are displayed in Table 3 in columns 3–7. The area of plots is displayed in



Ann Oper Res

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.

References

Alfandari, L., Lemalade, J. L., Nagih, A., & Plateau, G. (2009). A MIP flow model for crop-rotation planning

in a sustainable development context. Annals of Operation Research. doi:10.1007/s10479-009-0553-0.

Annetts, J. E., & Audsley, E. (2002). Multiple objective linear programming for environmental farm planning.The Journal of the Operational Research Society, 53, 933–943.

Barrett, C., Reardon, T., & Webb, P. (2001). Nonfarm income diversification and household livelihood strate-

gies in rural Africa: concepts, dynamics and policy implications. Food Policy, 26 (4), 315–331.

Beneke, R. R., & Winterboer, R. (1973). Linear programming applications to agriculture. Ames Iowa: IowaState University Press.

Blank, S. C. (1990). Returns to limited crop diversification. Western Journal of Agricultural Economics, 15,

204–212.

Blank, S. C., Carter, C. A., & McDonald, J. (1997). Is the market failing agricultural producers who wish to

manage risks? Contemporary Economic Policy, 15, 103–112.

Blank, S. C. (2001). Producers get squeezed up the farming food chain: a theory of crop portfolio composition

and land use. Review of Agricultural Economics, 23, 404–422.

Castellazzi, M. S., Wood, G. A., Burgess, P. J., Morris, J., Conrad, K. F., & Perry, J. N. (2008). A systematic

representation of crop rotations. Agricultural Systems, 97 (1–2), 26–33.

Collender, R. N. (1989). Estimation risk in farm planning under uncertainty. American Journal of Agricultural

Economics, 71, 996–1002.

Collins, R., & Barry, P. (1986). Risk analysis with single-index portfolio models: an application to farmplanning. American Journal of Agricultural Economics, 68, 152–161.

Detlefsen, N. K., & Jensen, A. L. (2007). Modelling optimal crop sequences using network flows. Agricul-

tural Systems, 94(2), 566–572.

Dogliotti, S., Rossing, W. A. H., & Van Ittersum, M. K. (2003). ROTAT, a tool for systematically generating

crop rotations. European Journal of Agronomy, 19(2), 239–250.

Ellis, F. (1998). Household strategies and rural livelihood diversification. Journal of Development Studies,35(1), 1–38.

http://dx.doi.org/10.1007/s10479-009-0553-0

http://dx.doi.org/10.1007/s10479-009-0553-0



Ann Oper Res

Ellis, F. (2000). Rural livelihoods and diversity in developing countries. Oxford: Oxford University Press.

El-Nazer, T., & McCarl, B. A. (1986). The choice of crop rotation: a modeling approach and case study.

American Journal of Agricultural Economics, 68(1), 127–136.

Fafchamps, M. (1992). Cash crop production, food price volatility, and rural market integration in the Third

World. American Journal of Agricultural Economics, 74(1), 90–99.

Figge, F. (2004). Bio-folio: applying portfolio theory to biodiversity. Biodiversity and Conservation, 13, 827–849.

Freund, R. J. (1956). The introduction of risk into a programming model. Econometrica, 24, 253–263.

Goland, C. (1993). Agricultural risk management through diversity: field scattering in Cuyo, Peru. Culture &

Agriculture, 45/46 , 8–13.

Haneveld, W. K., & Stegeman, A. W. (2005). Crop succession requirements in agricultural production plan-

ning. European Journal of Operations Research, 166 (2), 406–429.

Hardaker, J. B., Huirne, R. B. M., Anderson, J. R., & Lien, G. (2004). Coping with Risk in Agriculture (2nd

ed.). Oxfordshire: CABI Publishing.

Hazell, P. B. R. (1971). A linear alternative to quadratic and semivariance programming for farm planning

under uncertainty. American Journal of Agricultural Economics, 53, 53–62.

Hazell, P. B. R., & Norton, R. D. (1986). Mathematical Programming for Economic Analysis in Agriculture.

New York: Macmillan.Heady, Earl O. (1954). Simplified presentation and logical aspects of linear programming technique. Journal

of Farm Economics, 36 (5), 1035–1048.

Hoanh, C. T., Lai, N. X., Hoa, V. T. K., et al. (2000). Can Tho Province case study: land use planning under

the economic reform. In R. P. Roetter, H. Van Keulen, A. G. Laborte, H. H. Hoanh, & C. T. Van Laar

(Eds.), SysNet Research Paper Series: No. 3. Synthesis of methodology development and case studies

(pp. 47–52).

Johnson, S., Adams, R., & Perry, G. (1991). The on-farm costs of reducing groundwater pollution. American

Journal of Agricultural Economics, 73, 1063–1073.

Lazimy, R. (1982). Mixed-integer quadratic programming. Mathematical Programming, 23(1), 332–349.

Lewandrowski, J. K., & Brazee, R. J. (1993). Farm programs and climate change. Climatic Change, 23(1),

1–20.

Markowitz, H. M. (1952). Portfolio selection. The Journal of Finance, 7 (1), 77–91.Mimouni, M., Zekri, S., & Flichman, G. (2000). Modelling the trade-offs between farm income and the

reduction of erosion and nitrate pollution. Annals of Operation Research, 94(1–4), 91–103.

Nartea, G., & Bany, P. J. (1994). Risk efficiency and cost effects of geographic diversification. Review of

Agricultural Economics, 16 (3), 341–351.

Newbery, D. M. G., & Stiglitz, J. E. (1981). The theory of commodity price stabilization: a study in the

economics of risk . Oxford: Clarendon Press.

Nidumolu, U. B., van Keulen, H., Lubbers, M., & Mapfumo, A. (2007). Combining multiple goal linear

programming and inter-stakeholder communication matrix to generate land use planning options. Envi-

ronmental Modelling & Software, 22, 73–83.

Park, T. A., & Florkowski, W. J. (2003). Selection of peach varieties and the role of quality attributes. Journal

of Agricultural and Resource Economics, 28(1), 138–151.

Qiu, Z., Prato, T., & McCamley, F. (2001). Evaluating environmental risks using safety-first constraints.

American Journal of Agricultural Economics, 83(2), 402–413.

Radulescu, M., Radulescu, S., & Radulescu, C. Z. (2006). Mathematical models for optimal asset allocation.

Bucuresti: Editura Academiei Române (in Romanian)

Radulescu, M., Radulescu, S., & Radulescu, C. Z. (2009). Sustainable production technologies which take

into account environmental constraints. European Journal of Operational Research, 193, 730–740.

Radulescu, M., Radulescu, C. Z., Turek Rahoveanu, M., & Zbaganu, Gh. (2010). A portfolio theory approach

to fishery management. Studies in Informatics and Control, 19(3), 285–294.

Rafsnider, G. T., Driouchi, A., Tew, Bernard V., & Reid, D. W. (1993). Theory and application of risk anal-

ysis in an agricultural development setting: a Moroccan case study. European Review of Agricultural

Economics, 20, 471–489.

Reeves, L. H., & Lilieholm, R. J. (1993). Reducing financial risk in agroforestry planning: a case study inCosta Rica. Agroforestry Systems, 21, 169–175.

Roche, M. J., & McQuinn, K. (2004). Riskier product portfolio under decoupled payments. European Review

of Agricultural Economics, 31(2), 111–123.

Romero, C. (1976). Una aplicación del modelo de Markowitz a la selección de planes de variedades de

manzanos en la provincia de Lérida. Revista de Estudios Agro-sociales, 97 , 61–79.

Romero, C., & Rehman, T. (2003). Developments in Agricultural Economics: Vol. 11. Multiple criteria anal-

ysis for agricultural decisions (2nd ed.). Amsterdam: Elsevier.



Ann Oper Res

Romero, C. (2000). Risk programming for agricultural resource allocation: a multidimensional risk approach.

Annals of Operation Research, 94, 57–68.

Santos, L. M. R., Michelon, P., Arenales, M. N., & Santos, R. H. S. (2008). Crop rotation scheduling with

adjacency constraints. Annals of Operation Research. doi:10.1007/s10479-008-0478-z. Online First™.

Schaefer, K. C. (1992). A portfolio model for evaluating risk in economic development projects, with an

application to agriculture in Niger. Journal of Agricultural Economics, 43, 412–423.Teague, M. L., Bernardo, D. J., & Mapp, H. P. (1995). Farm-level economic analysis incorporating stochastic

environmental risk assessment. American Journal of Agricultural Economics, 77 (1), 8–19.

Turvey, C. G., Driver, H. C., & Baker, T. G. (1988). Systematic and nonsystematic risk in farm portfolio

selection. American Journal of Agricultural Economics, 70(4), 831–836.

Tziligakis, C. N. (1999). Relaxation and exact algorithms for solving mixed integer-quadratic optimization

problems. Thesis (S.M.), Massachusetts Institute of Technology, Sloan School of Management, Opera-

tions Research Center.

Vizvári, B., Lakner, Z., Csizmadia, Z., & Kovács, G. (2009). A stochastic programming and simula-

tion based analysis of the structure of production on the arable land. Annals of Operation Research.

doi:10.1007/s10479-009-0635-z.

Weintraub, A., Romero, C., Bjørndal, T., & Epstein, R. (Eds.) (2007). International Series in Operations

Research & Management Science: Vol. 99. Handbook of Operations Research in Natural Resources.Berlin: Springer.

Weintraub, A., Romero, C., Bjørndal, T., & Lane, D. (2001). Operational research models and the man-

agement of renewable natural resources: a review. Bergen Open Research Archive, Institute for Re-

search in Economics and Business Administration (SNF), Working Paper 11/2001. http://bora.nhh.no/

handle/2330/1371.

Wossink, G. A., de Koeijer, T. J., & Renkema, J. A. (1992). Environmental-economic policy assessment:

a farm economic approach. Agricultural Systems, 39, 421–438.

Zekri, S., & Herruzo, A. C. (1994). Complementary instruments to EEC nitrogen policy in non-sensitive

areas: a case study in southern Spain. Agricultural Systems, 46 , 245–255.

http://dx.doi.org/10.1007/s10479-008-0478-z

http://dx.doi.org/10.1007/s10479-009-0635-z

http://bora.nhh.no/handle/2330/1371




http://dx.doi.org/10.1007/s10479-009-0635-z

http://dx.doi.org/10.1007/s10479-008-0478-z

a portfolio theory approach

Documents