MULTIOBJECTIVE EVOLUTIONARY OPTIMIZATION OF

GREENHOUSE VEGETABLE CROP DISTRIBUTIONS

A. L. M´arquez

, F. Manzano-Agugliaro

, C. Gil

, R. Ca˜nero-Le´on

, F. G. Montoya

and R. Ba˜nos

Department of Computer Architecture and Electronics

Department of Rural Engineering

Department of Management

University of Almeria, Carretera de Sacramento S/N, Almeria, Spain

Keywords:

MultiObjective optimization, Greenhouse crop distribution, NSGA-II, msPESA, Risk management.

Abstract:

Multiobjective evolutionary algorithms (MOEAs) are known for their ability to optimize several objective

functions simultaneously to provide a representative set of the Pareto front, which is a set of problem solutions

representing a trade-off between the best values of each one of the objectives. This characteristic is specially

interesting for the optimization of many real world problems, such as the allocation of land resources to

maximize proﬁt while reducing the economical risks associated to different distributions of crops in southern

Spain, which has one of the largest concentrations of greenhouses in the world.

1 INTRODUCTION

While the purpose of any optimization procedure is

to ﬁnd the best possible solution to a certain problem,

many problems have several objectives to optimize

simultaneously. Therefore, the mission of Multi-

Objective Optimization Problems (MOPs) is to ﬁnd

trade-off solutions instead of a single one.

Since the formulation of the problem of maximiz-

ing proﬁt and reducing risks in crop area distribution

involves optimizing two conﬂicting objectives at the

same time, the aim is to obtain a set of solutions as

an approximation to the Pareto-optimal set (Fonseca

and Flemming, 1993). This set holds the best trade-

off solutions found by the problem solver. Gener-

ating the Pareto-optimal set in complex problems is

computationally expensive, and often infeasible, so a

large number of heuristic approaches such as simu-

lated annealing, tabu search, evolutionary algorithms,

memetic algorithms, etc. have been proposed in the

past.

In this study, the problem of planning greenhouse

crops has been approached from an economic point of

view with the aim of maximizing the proﬁt and min-

imizing the risk. Therefore, this paper evaluates the

performance of a multi-objective approach to solve

this optimization problem with two different algo-

rithms.

2 MULTI-OBJECTIVE

OPTIMIZATION: CONCEPTS

AND TECHNIQUES

This section introduces some multi-objective con-

cepts, that are of key importance to understand the

motivations and usefulness of the procedures pre-

sented on this paper. They are necessary because it

is impossible to exactly describe what a good approx-

imation to the Pareto Front is, in terms of a number

of criteria such as closeness to the Pareto set, diver-

sity, etc (Deb, 2002; Coello et al., 2002). Therefore,

there is a need to introduce some Multi-Objective Op-

timization concepts:

Deﬁnition 1. Multi-objective Optimization is the

process of searching for one or more decision vari-

ables that simultaneously satisfy all constraints, and

optimize an objective function vector that maps the

decision variables to two or more objectives.

minimize/maximize(f

(s)), ∀k ∈ [1,K]

Each decision vector or solution s={(s

, s

, .., s

)}

represents accurate numerical qualities for a MOP.

The set of all decision vectors constitutes the decision

space. The set of decision vectors that simultaneously

satisﬁes all the constraints is called feasible set (F).

218

Márquez A., Manzano-Agugliaro F., Gil C., Cañero-León R., Montoya F. and Baños R. (2009).

MULTIOBJECTIVE EVOLUTIONARY OPTIMIZATION OF GREENHOUSE VEGETABLE CROP DISTRIBUTIONS.

In Proceedings of the International Joint Conference on Computational Intelligence, pages 218-223

DOI: 10.5220/0002317002180223

 SciTePress

Figure 1: Pareto-dominance relations in a two-objective problem.

The objective function vector (f) maps the decision

vectors from the decision space into a K-dimensional

objective space Z∈ℜ

, z=f (s), f(s)={ f

(s), f

(s),...,

(s)}, z∈Z, s∈F.

Given a MOP with K≥2 objectives, instead of giv-

ing a scalar value to each solution, a partial order is

deﬁned according to Pareto-dominance relations, as

we detail below.

Deﬁnition 2. Order Relation between Decision

Vectors. Let s and s’ be two decision vectors. The

dominance relations in a minimization problem are:

s dominates s’ (s≺s’) iff

(s)< f

(s’) ∧ f

′

(s)6> f

′

(s’), ∀k’6=k∈[1,K]

s, s’ are incomparable (s∼s’) iff

(s)< f

(s’) ∧ f

′

(s)> f

′

(s’), k’6=k∈[1,K]

Deﬁnition 3. Pareto-optimal Solution. A solution

s is called Pareto-optimal if there is no other s’∈F,

such that f(s’)< f(s). All the Pareto-optimal solutions

deﬁne the Pareto-optimal set.

Deﬁnition 4. Non-dominated Solution. A solution

s∈F is non-dominated with respect to a set S

′

∈F if

and only if 6 ∃s’∈S

′

, verifying that s

′

≺s.

Deﬁnition 5. Non-dominated Set. Given a set

of solutions S

′

, such that S

′

∈F and Z

′

=f(S

′

), the

function ND(S

′

) returns the set of non-dominated

solutions from S

′

ND(S

′

) = {∀s

′

∈ S

′

is non-dominated by any other

s”,s” ∈ S’}

Figure 1 graphically describes the Pareto-

dominance concept for a minimization problem with

two objectives (f

and f

). Figure 1(a) shows the loca-

tion of several solutions. The ﬁlled circles represent

non-dominated solutions, while the non-ﬁlled ones

symbolize dominated solutions. Figure 1(b) shows

the relative distribution of the solutions in reference

to a solution s. There exist solutions that are worse

(in both objectives) than s, better (in both objectives)

than s, and incomparable (better in one objective and

worse in the other).

2.1 NSGA-II

The ﬁrst algorithm we have used is the Non-

dominated Sorting Genetic Algorithm II (Deb et al.,

2000) which is an extended version of NSGA

(Srinivas and Deb, 1994). NSGA-II creates a

non-dominance based hierarchy by using its non-

dominated sorting procedure, which is combined with

a crowded comparison operator that keeps diversity

by using the euclidean distance between solutions to

specify the ﬁtness of each one. In its operation (see

(Deb et al., 2000)) non-dominatedsolutions are added

to the external archive by using the dominance hier-

archy previously created, until it is completely ﬁlled.

The remaining solutions are discarded.

2.2 msPESA

The second algorithm we have used is msPESA (Gil

et al., 2007), a hybrid algorithm that combines aspects

of PESA (Corne et al., 2000) and NSGA-II. It uses a

small internal population and a larger external popu-

lation (archive), implements a variant of the archive-

update strategy of PESA, and takes some ideas of the

selection mechanism used in NSGA-II.

When a candidate solution enters the archive, in-

stead of removing the solutions it dominates from the

archive, msPESA keeps them for the sake of improv-

ing genetic variability. It can perform a local search

procedure using the mutation operator over the candi-

date solution, to try to ﬁnd more good solutions in its

neighbourhood. After that, the non-dominated solu-

tions generated from the local search are stored within

the archive. Once the archive is full, the squeeze fac-

tor is used (see below) to remove the excess of solu-

tions from the archive.

MULTIOBJECTIVE EVOLUTIONARY OPTIMIZATION OF GREENHOUSE VEGETABLE CROP DISTRIBUTIONS

219

The squeeze factor is calculated with the help of

the hyper-grid strategy. In msPESA the objective

space is divided in a (N − 1) dimensional grid, which

can be used to achieve a ﬁner resolution with the same

memory requirements. It even allows using as many

divisions of the search space as subjects can be stored

into the Archive, therefore allowing the procedure to

obtain an evenly distributed Pareto front, since it tries

to keep a maximum of one subject on each cell of the

hyper-grid.

3 OPTIMIZING CROP

DISTRIBUTION

Plastic-covered greenhouses have undergone rapid

expansion in recent years, covering a surface of over

1.600.000ha worldwide (Espi et al., 2006), which are

mainly distributed in two geographical areas: The Far

East (whose maximum contributors are China, Japan

and Korea) with almost 80% of the surface (Jiang

et al., 2004) and the Mediterranean area with about

a 15% of the world’s greenhouse-covered extensions

(Pardossi et al., 2004).

Table 1: Main economical data obtained from the different

greenhouse crop options (e/m

2004-2005 2005-2006

Crop Revenue Cost Revenue Cost

Pe 3.85e 1.70e 5.62e 2.12e

Pe-M 5.35e 2.27e 4.43e 2.82e

Pe-W 5.55e 2.51e 4.48e 4.02e

T-M 6.90e 2.23e 6.18e 3.76e

T-W 6.69e 1.99e 6.14e 2.56e

T-T 9.07e 3.91e 8.21e 4.88e

GB-GB 5.40e 2.68e 8.90e 2.74e

GB-M 5.17e 2.06e 6.16e 1.98e

GB-W 3.55e 1.89e 6.20e 2.18e

Cu 10.63e 1.88e 5.83e 2.58e

Cu-M 6.80e 1.92e 4.64e 2.53e

Cu-W 7.01e 2.16e 4.69e 2.73e

Cu-GB 7.03e 2.53e 7.14e 3.09e

Co-Co 8.58e 2.14e 5.12e 2.22e

Co-W 5.06e 1.79e 6.29e 2.03e

Co-M 4.86e 1.55e 6.24e 1.83e

The greenhouse crop surface at the province of

Almeria (south-eastern Spain) is about 30.000ha with

an estimated production of 3x10

kg of produce at

an approximate value of 1,384· 10

e (IEC, 2009),

where 80% of the cultivated crop varieties are: pep-

per, tomato, green beans, cuccumber, courgette, wa-

termelon and melon (Manzano-Agugliaro, 2007).

The distributions of these crop surfaces change every

year, sometimes causing a decrease in average prices

compared with the previous year. The behaviour of

average prizes is inversely proportional to the quanti-

ties produced (IEC, 2009).

In previous works, weighted goal programming

using utility functions has been used as a method-

ology for the analysis and simulation of agricul-

tural systems (Sumpsi et al., 1997; Amador et al.,

1998). These techniques have been used in a

decision-making process in planning crops (Berbel

and Gomez-Limon, 1997). This is consistent with

how classical optimization methods suggested con-

verting the multi-objective problems into a single-

objective formulation by combining the objectives in

only one mathematical function (Fonseca and Flem-

ming, 1993).

3.1 Problem Information

The seven main crop varieties mentioned above are

combined in sixteen vegetable crop alternatives that

are the object of this study. The data fed to the solver

has been obtained from an accountancy tracking of

46 and 49 greenhouses in 2004-2005 and 2005-2006

seasons respectively. There is additional information

for each crop to be dealt with, such as production,

revenue, fertilisers, needed water per m

, or ﬁxed

costs (depreciation, soil disinfection, etc). With this

information we can obtain the Gross Margin (GM)

(see table 1) for each crop option and each year (rev-

enue minus variable costs), irrigation water consump-

tion (m

), nitrate consumption (Fertilizing unit

per m

). For the actual crop distributions we have ob-

served a gross margin of 84,666e and risk value of

465· 10

3.2 Objectives to Optimize

There are sixteen variables in this problem which cor-

respond to the surface distribution for each one of the

sixteen crop alternatives mentioned before, there are

two restrictions and two main objectives to optimize

that are of the highest interest to farmers:

Maximizing Proﬁt. To do this, the gross margin

(GM) is maximized for the various options pro-

posed:

GM =

∑

i=1

(GM

· X

) (1)

Where GM

is the gross margin of option i per sur-

face area unit (e/m

) and X

is the surface area

that the crop alternative covers.

Minimizing Risk. Since agricultural production

may be affected by random conditions, the risk

is a factor that inﬂuences the choices of the

crop options. To calculate risk, the variance and

IJCCI 2009 - International Joint Conference on Computational Intelligence

220

Figure 2: Pareto front of Gross Margin (GM) vs. Risk for NSGA-II and msPESA.

covariance matrix is used for the gross margins

of the different crop options, based on the above

data.

R = X

[cov]X

(2)

Since these objectives are on conﬂict, the use of

Multi-Objective Evolutionary Algorithms for gener-

ating a Pareto Front with the distributions of the crop

alternatives seems appropriate, especially considering

that with enough abundance of widely distributed so-

lutions over the objective space we could offer the

greenhouse owners the choice to select the most ap-

propriate distribution of crop alternatives considering

his ﬁnancial needs or the risks that the farmer is able

to withstand.

There are also a couple of constraints that are to

be taken into account:

• The total surface area is limited to 2.5ha, which is

the average greenhouse surface.

∑

i=1

= 25,000m

(3)

• The maximum surface area for a certain crop

should never be higher than the 40% of the total

surface as a restriction, because it produces a mar-

ket ﬂood of the product, leading to an important

drop in the prices, therefore reducing the revenues

for that crop to the point that it could even gen-

erate a money loss to the owners of the affected

lands.

3.3 Evolutionary Operations

A ﬂoating point numeric representation for each one

of the sixteen variables has been chosen, because it is

the most natural representation for this problem. Fur-

thermore, it is necessary to implement speciﬁc opera-

tors in MOEAs to be able to work with this problem,

such as mutation, crossover and a chromosome repair

procedure. While in other problems there might not

arise such need, it is necessary to implement a chro-

mosome repair mechanism for the crop surface opti-

mization procedureto guarantee that the generated so-

lutions fall within the problem restrictions, therefore

making them feasible to use in the next generation of

solutions for each iteration of the MOEA.

Mutation. The mutation operation is the simplest of

all three, since it just iterates through all the chro-

mosome variables applying a random change of

up to a ±25% of its initial value.

Crossover. We applied a multipoint chromosome

crossover, where each one of the values of the

genotype for each parent chromosome has a 50%

of probabilities to ending up as part of the geno-

type for the chromosome of the next generation.

Chromosome Repair Mechanism. Its function is to

normalize the total surface represented by each

one of the chromosome variables to a surface of

2.5ha. Once it has been done, then we proceed to

calculate the total surface belonging to each one of

MULTIOBJECTIVE EVOLUTIONARY OPTIMIZATION OF GREENHOUSE VEGETABLE CROP DISTRIBUTIONS

221

Figure 3: Comparison between surfaces on the extreme opposite solutions in the Pareto Front for a single run of NSGA-II.

the seven crops, to check if any of them extends

over a 40% of the total surface. If that is the sit-

uation, we proceed to proportionally redistribute

the excess surface between the rest of the crops.

This means that if, for example, the surface of cu-

cumber is distributed between 3 crop alternatives,

each one of them would be substracted a propor-

tional amount of the excess surface, meaning that

the greatest contributor to the situation is the al-

ternative that will experience a greater surface re-

duction. This also ensures that the crop reduction

is not greater than the available surface for the af-

fected alternative.

4 EXPERIMENTAL RESULTS

Since Multi-Objective Evolutionary Algorithms are

highly conﬁgurable procedures (Goldberg, 1989), we

have chosen to use similar parameters for both of

them, always considering the particularities of each

alternative. For both algorithms we chose to execute

the same number of function evaluations (20,000) to

allow them to perform enough evaluations to gener-

ate good solutions. The parameters chosen for each

procedure are:

NSGA-II. This algorithm works at its best when both

the internal and external population have the same

size.

• Population and External population: 100 indi-

viduals.

• Mutation rate: 0.1

msPESA. For msPESA, working with a small inter-

nal population allows the procedure to generate a

greater number of generations. Since all the best

solutions are always kept in the archive, this al-

lows it to generate a good approximation Pareto

Front.

• Population: 10 individuals.

• External Population (Archive): 100 individu-

als.

• Mutation rate: 1.0

The ﬁgure 2 represents a typical execution of

NSGA-II and msPESA for the problem at hand. As

we can observe, this graphic shows how the non-

dominated solutions obtained as approximation to

the Pareto-optimal front have a clear deﬁnite shape,

meaning that there is a strong relation between risk

and the gross margin, and that they are opposite objec-

tives, since the ideal situation would yield a maximum

gross margin with a low risk. It is worth noting that

both algorithms yield similar results, being NSGA-II

the algorithm that generates the best Pareto-optimal

set of solutions using the proposed experimental pa-

rameters.

IJCCI 2009 - International Joint Conference on Computational Intelligence

222

5 CONCLUSIONS

The nature of the problem of crop surface optimiza-

tion allows it to be efﬁciently represented and devel-

oped as a Multi-Objective problem that can be solved

using any of the current algorithms. This allows the

user to search and analyze a wide range of possible

situations to choose the simulated solution that may

be of the best interest for the situation of a particular

farmer.

As ﬁgure 3 shows, the differences between a

higher proﬁt-higher risk simulation and a lower proﬁt-

lower risk one are signiﬁcant, meaning that green

beans are a relatively safe value to use, while cucum-

ber is the most proﬁtable crop to plant, but its associ-

ated risks are high enough to be considered.

The new distribution of crops obtained with this

method, shows better Gross Margin and lower Risk in

the minimum point of the Pareto Front than the real

situation. This means that the crop distributions may

be optimized in order to maximize the beneﬁts for the

greenhouse farmers.

ACKNOWLEDGEMENTS

This work has been ﬁnanced by the Spanish Ministry

of Innovation and Science (TIN2005-00447), Min-

istry of Science and Technology (AGL2002-04251-

C03-03) and the Excellence Project of Junta de An-

daluc´ıa (P07-TIC02988), in part ﬁnanced by the Eu-

ropean Regional Development Fund (ERDF).

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