A GENETIC ALGORITHM FOR CROP ROTATION

Angelo Aliano Filho

, Helenice de Oliveira Florentino

and Margarida Vaz Pato

Curso de Biometria, Departamento de Bioestat

ıstica, IB, UNESP, 18618-970, Botucatu, SP, Brazil

Departamento de Bioestat

ıstica, IB, UNESP, 18618-970, Botucatu, SP, Brazil

CIO- FCUL and ISEG - UTL, 1200-781, Lisboa, Portugal

Keywords:

Optimization, Genetic algorithm, Crop rotation.

Abstract:

In the last few years, crop rotation has gained attention due to its economic, environmental and social impor-

tance which explains why it can be highly beneﬁcial for farmers. This paper presents a mathematical model

for the Crop Rotation Problem (CRP) that was adapted from literature for this highly complex combinatorial

problem. The CRP is devised to ﬁnd a vegetable planting program that takes into account green fertilization

restrictions, the set-aside period, planting restrictions for neighboring lots and for crop sequencing, demand

constraints, while, at the same time, maximizing the proﬁtability of the planted area. The main aim of this

study is to develop a genetic algorithm and test it in a real context. The genetic algorithm involves a cons-

tructive heuristic to build the initial population and the operators of crossover, mutation, migration and elitism.

The computational experiment was performed for a medium dimension real planting area with 16 lots, con-

sidering 29 crops of 10 different botanical families and a two-year planting rotation. Results showed that

the algorithm determined feasible solutions in a reasonable computational time, thus proving its efﬁcacy for

dealing with this practical application.

1 INTRODUCTION

Brazil has an agricultural tradition, and a variety of

grains and vegetables are planted on a large scale.

Natural conditions, such as climate, fertile soil and

relief, foster the development of this important eco-

nomic sector.

(Altieri, 2002) and (Gliessman, 2000) mention

that conventional agricultural production is mainly

based on monoculture given the fact that its oper-

ational cost is low and can be more easily imple-

mented. But this leads to several harmful factors,

such as the extensive use of capital and toxic pesti-

cides that are extremely damaging to the environment

and facilitates the action and prevalence of pests and

pathogens. Other negative economic aspects arising

from this practice are the impoverishment of small

farmers and the decreasing volumes of certain agri-

cultural products.

To prevent these drawbacks, polyculture should

be expanded, in a sustainable fashion, besides consi-

dering the environmental conservation of planting a-

reas, pest control (and thus the less intensive use of

chemical fertilizers) and economic gain.

In the sequel, crop rotation has been object of

study within the agricultural and ecological ﬁelds.

The basic idea consists in annually alternating diffe-

rent plant families within the same agricultural area.

The selected species should have commercial and

soil-recovery purposes. If crop rotation is properly

adopted and conducted for a sufﬁciently long pe-

riod, its advantages are numerous. A practice such

as this improves the physical, chemical and biologi-

cal characteristics of soil, helps disease and pest con-

trol, replaces organic matter, protects the soil from cli-

mactic agents and, in addition, provides a diversiﬁed

production of food and other products.

Crop rotation has also been studied within the ope-

rational research domain, namely by (Lemalade et al.,

2011), (Santos, 2009) and (Santos et al., 2011). In this

paper we propose a new method to tackle the Crop

Rotation Problem (CRP) based on a genetic algorithm

(GA) which embeds a constructive heuristic.

2 MATHEMATICAL MODEL

In the mathematical model proposed for the CRP by

(Santos et al., 2011) the objective is to maximize oc-

cupation of a speciﬁc planting area and the follo-

454

Aliano Filho A., de Oliveira Florentino H. and Vaz Pato M..

A GENETIC ALGORITHM FOR CROP ROTATION.

DOI: 10.5220/0003761904540457

In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 454-457

ISBN: 978-989-8425-97-3

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

wing rotation constraints are taken into account for

the planning horizon:

(a) Sowing Season: need to respect the time of plan-

ting and the life cycle of each culture;

(b) Continuity for Same-family Crops: plants belon-

ging to the same family must not be planted in

adjacent parcels of land or lots;

same family must not be planted consecutively on

the same lot;

(d) Green Fertilization: in the planning horizon each

lot must have a plant of the leguminosae family

and be subject to the above mentioned conditions

(a) and (b), also each of these must be planted only

once;

(e) Set-aside Period: a set-aside period must be

scheduled on each lot.

In this study, the objective of the CRP is to maxi-

mize proﬁtability and a new type of restraint is im-

posed:

(f) demand - each culture has a pre-stablished market

demand must be satisﬁed.

The planning horizon divided into M periods of

similar duration, a set of N crops belonging to N

plant families and the planting area with L lots are

taken into account. Other parameters follow:

• C: set of trade crops;

• A: set of crops for green fertilization;

• F

: set of plants of the p family, p = 1..N

;

• t

: crop planting cycle i, including soil preparation

and harvesting;

• l

i j

: proﬁtability of crop i in the period j per unit

of area;

• I

= [C

]: crop planting interval i, in which C

the earlier period and T

is the later period.

• p

i j

: production of crop i in the period j per unit

of area;

• D

: demand for crop i;

• I

= [C

]: demand interval of crop i, in which

is the earlier period and T

is the later period

of the demand;

• S

: set of lots adjacent to lot k;

• area

: area of lot k.

A binary linear programming model is described

below:

maximize z =

∑

i∈C

∑

j∈I

∑

k=1

area

i j

i jk

(1)

subject to

∑

i∈F

−1

∑

r=0

∑

v∈S

i( j−r)v

≤ L

1 −

∑

i∈F

−1

∑

r=0

i jk

p = 1..N

, j = 1..M,k = 1..L (2)

∑

i∈F

∑

r=0

i( j−r)k

≤ 1, p = 1..N

, j = 1..M,k = 1..L (3)

N+1

∑

i=1

−1

∑

r=0

i( j−r)k

≤ 1, j = 1..M, k = 1..L (4)

∑

i∈A

∑

j=1

i jk

≥ 1, k = 1..L (5)

∑

j=1

n jk

≥ 1, k = 1..L (6)

∑

j ∈I

∑

k=1

area

i jk

≥ D

, i ∈ C (7)

i jk

∈ {0,1}, i = 1..N + 1, j ∈ I

, k = 1..L. (8)

where, for convenience of notation, the set-aside pe-

riod is represented by crop (N + 1) and if j − r ≤ 0

then j −r is replaced by j − r + M. Here, the decision

variable x

i jk

is equal to 1 if crop i is planted in pe-

riod j (eventually continues in the following periods

according to its cycle) in lot k, and 0 otherwise.

In the proposed model, the deciding variable x

i jk

will be 1 if crop i is in its planting period initiated

in period j and if it is planted in parcel k, and 0 if

otherwise.

The objective function (1) sets out to maximize

the proﬁtability of the rotation performed on the given

area. Constraints (2) prevent the plants of the same

family from being neighbors. Constraints (3) for-

bid the same plant family from being consecutively

planted on the same lot k. Constraints (4) prevents

two plants from occupying the same lot in the same

time interval. This means that, if crop i is planted

on lot k, t

periods must elapse before a new plant

occupies the same space. Constraints (5) and (6) en-

sure that each lot has at least one green fertilization

application and a set-aside period, respectively. No-

tice that the neighborhood and consecutive planting

restrictions do not apply to the set-aside period. Fi-

nally, constraints (7) impose satisfaction of demand

for crops during the respective period.

A similar objective function (1) and the same

constraints (2) to (6) and (8) are shared by this model

with the one proposed by (Santos et al., 2011).

A GENETIC ALGORITHM FOR CROP ROTATION

455

3 METHODOLOGY

Due to the high computational resources that are nec-

essary to exactly solve the CRP instances arising from

real applications, non-exact approaches are advisable

for this problem. Genetic algorithms are specially

adapted to deal with this type of combinatorial opti-

mization problem insofar as they can easily deal with

the optimization objective and the many constraints

involved. Hence, a new GA was developed for the

CRP.

The above model (1) to (8) uses binary variables,

but it is more convenient for this algorithm to be based

on integer decision variables taking values in the in-

terval [1,N + 1]. Otherwise, modeling with binary

variables would result in bigger dimension chromo-

somes and enhanced difﬁculties in dealing with the

constraints. One solution to the problem is associated

with an individual identiﬁed by a single chromosome,

which is encoded through an L × M integer matrix.

Its element (k, j) belongs to [1, N + 1] and identiﬁes

which crop is being planted on lot k and in period j,

for all k = 1..L and j = 1..M.

With the purpose of exploring the set of solutions

more efﬁciently, the initial population for the GA was

determined by a new constructive heuristic. Each ini-

tial solution/individual is built lot by lot thus imposing

the planting sequence constraint on the same lot, that

is, rotation conditions (a) and (b).

The ﬁtness of an individual is initially set equal to

the sum of the proﬁtabilities on all lots. Then an expo-

nential penalization process is used to punish the in-

feasible solutions due to violation of the remaining ro-

tation conditions. Hence, penalizations of the neigh-

borhood, set-aside period, green fertilization and de-

mand violations are added. Considering that their

sum is p, the ﬁtness of the individual is multiplied

by e

−p/K

, where K is a positive constant equal to 10.

The GA was applied with 120 generations, 451 in-

dividuals (solutions) in the population of all the gen-

erations and using selection, crossover, mutation, mi-

gration and elitism operators.

The selection method used was a biased roulette

wheel.

As to crossover the option favored a uniform

crossover with a rate of 80%. The chromosome break

points for this genetic operator were horizontally and

randomly selected (to keep feasibility in the line).

Mutation consists of randomly selecting lots of the

individuals and replacing the respective cultures with

crops constructively assigned.

The migration operator was devised to avoid pre-

mature convergence of the algorithm and, at the same

time, it explores the feasible region more efﬁciently.

In this context, 50% of the population in the genera-

tions 120γ is replaced by randomly generated indi-

viduals (with γ, a control parameter taking the value

0.80).

Migration as well as mutation rates both equal to

5%.

The elitism consists of saving the best solution be-

fore the action of the operators in each generation and

inserting it in the population for the next generation.

4 COMPUTATIONAL RESULTS

The computational experiment was performed with

the algorithms coded using Matlab software, version

7.4.0 R2007a, on Core 2 Quad microcomputers with

2GB memory and 250 GB hard-disk memory.

The computational experiment took into account a

real instance of CRP with a planting area that presents

16 non-parallel lots, as shown is Figure 1, and other

real data for crops coming from a medium size plan-

ting area at S

ao Paulo state, in Brazil.

Figure 1: Plantation area.

A two-year planting program was considered and

each period ﬁxed at one month. Crops 23 to 29 were

selected for green fertilization and crop 30 for the set-

aside period. The plants represented by 1 to 25 corres-

pond to marketing purposes.

The above GA ran 100 times for this real life pro-

blem.

To access the performance of the GA in this con-

text, we measured the average CPU time per GA run

(t in seconds), the number of times the algorithm pro-

vided a feasible best solution out of 100 (α), the ave-

rage proﬁtability of the best solution found per run (z

in R$), the average penalizations for the ﬁtness of the

best solution at the initial population per run (p

inicial

)

and the same for the best solution at the ﬁnal popu-

lation per run (p

f inal

). Lastly, the value ∆z repre-

sents the average relative deviation between the proﬁ-

tability of the best ﬁrst and best ﬁnal solutions found

at each run of the GA. These last three ﬁgures are de-

termined with the purpose of accessing the improve-

ment achieved with the algorithm from its beginning

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

456

Figure 2: The best rotation for the real instance with 16 lots.

to the end. These values are displayed in Table 1.

Table 1: Mean of computational results.

t z α p

inicial

f inal

∆z

1,017 1.53 ×10

72 44.0 0.43 0.23

From the results in Table 1 one may observe that

the 100 runs of the GA consumed 1017 seconds of

CPU time. However, for 72 of the 100 GA runs

the best solution attained was feasible. The aver-

age penalizations for the ﬁtness of the ﬁrst genera-

tion best solution are high (p

inicial

= 44.0) at the end

the best solution ﬁtness is penalized on average with

f inal

= 0.43. It means that the best solution from

most of the 100 GA runs is non-penalized, that is, it

is feasible, thus satisfying all the CRP constraints.

The ﬁgure ∆z = 0.23 represents an average in-

crease of 23% on the total proﬁtability of the best so-

lution in the population from the beginning to the end

of the algorithm.

The best feasible solution determined by the GA

following the 100 runs, a rotation for the planting area

studied, is given in Figure 2.

5 SOME CONCLUSIONS

In this study, a new GA was deve loped, along with

a simple constructive heuristic, for the Crop Rotation

Problem.

The method provided good quality feasible solu-

tions in short computing time, thus proving to be a vi-

able, simple and efﬁcient approach to tackling a real

instance of this problem of a highly complex combi-

natorial nature.

Moreover, the model presented for the CRP pos-

sesses perfect real applicability insofar as it takes into

account both technical as well as economic consider-

ations.

Therefore, the methodology is a promising tool to

help farmers in decision-making processes.

The authors intend to continue this study by tes-

ting the methodology with more real cases besides

working with instances randomly built to simulate the

real contexts addressed.

ACKNOWLEDGEMENTS

We wish to thank CAPES (Coordenac¸

Aperfeic¸oamento Pessoal de N

ıvel Superior) ﬁnan-

cial support by the ﬁrst author, FAPESP (Fundac¸

de Amparo

a Pesquisa do Estado de S

ao Paulo,

Grant No. 2009/14901-4, Brazil), FUNDUNESP

(Fundac¸

ao para o Desenvolvimento da UNESP,

Brazil), PROPG UNESP for their ﬁnancial support.

The third author’s research is partially supported

by the Foundation for Science and Technology of

Portugal, under the project POCTI/ISFL/152. The

authors are grateful to Sr. Sebasti

ao Soares de

Oliveira, owner of a farm at Santa Cruz de Rio Pardo

for kindly providing us with all the real data for the

study.

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