Problem description and model formulation
Assuming that crop demands is the total demands for the whole period; plots are homogeneous,
thus the productivity of fertilizer is the same;
The plan must be sustainable for over the long term not just for one year.
For agricultural production, a vegetable rotation is a planting calendar presenting how to
arrange the sequence of vegetables on a plot in the given time horizon. The problem can be
described as follows:
A given set of C vegetables C1 C = { ，2，..., } is to be cultivated on a given set of H
plots H 12 H = { ，，..., } within a predetermined duration T of T periods, T 12 T = , { ，... } .
The H plots are assumed to be homogeneous. Each vegetable i ,∈i C , is planting
repeatable and has a production duration i t t I ∈ including the time for preparing the soil and
harvesting. In addition, all of the crops are assumed to have only one harvest occurring in the
last period of its production duration. The fixed production cost of vegetable i ,∈i C , on plot
h without using chemical inputs at the period t is ith , , c , and this cost is charged at the period
vegetable i planted. The vegetable i , ∈i C , belongs to F botanic families,
F 12 F = , ,..., { }, and two vegetables from the same botanic family cannot be cultivated on
one plot immediately one after the other. Besides, vegetables from the same botanic family also
cannot be cultivated at the same time on adjacent plots.
For example: There are four vegetables C1, C2, C3 and C4 with different planting date
and production duration. C1 and C3 are from brassicaceae botanic family, C2 and C4 are from
solanaceae botanic family and cucurbitaceae botanic family, respectively. Green manure and
fallow period are takes as two special vegetables with one month production duration and all
year round planting date. Fig. 1 shows a sustainable vegetable rotation schedules for two
adjacent plots. For plot 1, fallow period is planted in the first month, C1, green manure and C3
are orderly planted. For plot 2, C2 is cultivated in the first month and harvested in the fifth
month, C4, fallow period and green manure are orderly planted after that. The vegetable rotation
schedules for plot 1 and plot 2 satisfy the constraints of (a)-(c).
Plot 1 F C1 G C3
Plot 2 C2 C4 F G
1 2 3 4 5 6 7 8 9 10 11 12
Fig.1. Sustainable vegetable rotation schedules for plot 1 and plot 2
Due to the increasing concerns about sustainable development in agricultural production,
green manures and fallow periods are suggested to be respected in each rotation period (Santos
et al., 2010; Santos et al., 20111
; Santos et al., 2015; Alfandari et al., 2014). In the model, U
unit green manures iG C1 G ∈ = + ,..., { } and V unit fallow periods must be arranged in
each rotation on plot h H ∈ . Green manures and fallow period are taken as special vegetables.
The production duration of green manures and fallow period is assumed to one period,
indicating that i t 1 = for i C1 C G1 ∈ + ++ { ,..., }. The set of vegetablesC1 C = { ，2，..., },
green manures G C1 G = + ,..., { } , and fallow periods all belong to the set of
1 Santos Lana Mara R. dos, Philippe Michelon, Marcos Nereu Arenales, Ricardo Henrique Silva Santos.
Crop rotation scheduling with adjacency constraints, Annals of Operation Research, 2011,190:165-180.
1
I 12 C G 1 = ..., + { ，， + } . If vegetable i ,∈i I , is planted on plot h at period t T ∈ , decision
variable ith , , x takes the value 1, otherwise, takes the value zero.
In addition, chemical input j J 12 J ∈ = , ,..., { } (e.g. fertilizer/pesticide) plays a vital role
in vegetable growth. Under desirable level, unit chemical input j can boost additional output
of vegetable i ,∈i C , on unit area of plot h by i jh , , a units. However, if the quantity of
chemical inputs exceeds the maximum admissible level, it will result in many side-effects such
as water pollution, damage to human health (Santos et al., 2010; Radulescu et al., 20142
), soil
contamination and emission of NO2 into the air (Wu, 20113
). Thus, in our model, the quantity
of chemical inputs j for vegetable i cannot exceeds the maximum admissible level of Li j , and
cost j cfp will be charged for buying unit chemical input j . Green manures and fallow period
do not need any chemical inputs, while fixed cost i h, cg will be charged for
i C1 C G1 ∈ + ++ { ,..., } per unit area of plot h .
In practice, the total production costs in each rotation must be smaller than budget B , and
the production quantity must be meet the demand Di .
Notation
i Indexes of vegetables, green manures and fallow period, where i C = , ... 1 2，， denotes
vegetables, C is the set of all vegetables; iC 1 G = ,..., + denotes green manure, G
is the set of green manures, where I G = , ..., {1 2， }.
j Index of chemical inputs, where j= , ... 1 2， , j J ∈ , where J is the set of chemical
inputs.
t Index of periods, where t= , ... 1 2， ,t T ∈ , whereT is the set of time periods.
h Index of plots, where h =1 2, ,..., h H ∈ , where H is the set of plots.
Parameters
F p( ) Set of botanic family, p N ∈ , where N denotes the number of botanic family, and
N C ≤ .
i S Set of time periods in which vegetable i can be cultivated.
ith a Quantity of vegetable i produced on unit area of plot h at period t without chemical
inputs, where i C ∈ ,t T ∈ and h H ∈ .
ijh b Quantity of vegetable i produced on unit area of plot h with unit chemical input j ,
where i C ∈ , j J ∈ and h H ∈ .
ith c Production cost of vegetable i on plot h at the period t without chemical inputs,
where i I ∈ , t T ∈ and h H ∈ .
2 Radulescu Marius, Constanta Zoie Radulescu, Gheorghita Zbaganu. A portfolio theory approach to crop
planning under environmental constrains, Annals of Operation Research, 2014,219:243-264.
3 Wu Yanrui. Chemical fertilizer use efficiency and its determinants in China’s farming sector. China
Agricultural Economic Review, 2011, 3: 117-130.
2
j cfp Cost of unit chemical input j , where j J ∈ .
it p Price of unit vegetable i at period t , where i C ∈ and t T ∈ .
i pd Production duration of vegetable i from preparing the land to being harvest, where
i C ∈ .
Di Demand of vegetable i , where i C ∈ .
B Available budget for each rotation.
Lij Maximum admissible level of chemical input j for vegetable i , where i C ∈ , and
j J ∈ .
M A large number.
tg Minimum time delays for green manures and fallows.
h h, ' e If plot h and plot h' are adjacent, h h, ' e takes the value of one, otherwise taking the
value of zero. The value of h h, ' e depends on the distribution of plots.
Decision variables
ijth y Quantity of chemical input j applied to vegetable i on plot h at period t , where
i C ∈ , j J ∈ , t T ∈ and h H ∈ .
ith x ith x = 1if vegetable, green manure or fallow period i is planted on plot h at period t ;
ith x = 0 otherwise. Where i I ∈ , t T ∈ and h H ∈ .
ith z Production quantity of vegetable i on unit area of plot h at period t , where i C ∈ ,
t T ∈ and h H ∈ .
Model formulation
Max it ith j itjh ith ith
iCtTh H j J i I tTh H
p z cfp y c x ∈∈∈ ∈ ∈∈ ∈
∑∑∑ ∑ ∑∑∑
subject to:
ith ith+ j ijth
i I tTh H iC j JtTh H
c x cfp y B ∈∈ ∈ ∈ ∈∈ ∈
∑∑∑ ∑∑∑∑ ≤ (1)
( ) i
i
t pd 1
i t pd h ith ith ijh ijth
t t j J
z ax by
= ∈
= ∑ ∑+ , , i ∈ ∈ iC tS and ∈h H (2)
, , i i t pd h ith z Mx + 1 ≤ , , i ∈ ∈ iC tS , i t pd T + ≤ and ∈h H (3)
ith i
tTh H
z D ∈ ∈
∑ ∑ ≥ , , i ?∈ ?∈ iC tS , i t pd T + ≤ and? ∈h H (4)
i t pd 1
ijt h ith
jJ t t
y Mx
∈ =′
∑ ∑ ≤ ,∈i C , ,i i∈ + ≤ t S t pd T , and ∈h H (5)
ijth ij y L ≤ ,∈ ∈ iC jJ , , ∈t T and ∈h H (6)
ith
i I
x ∈
∑ ≤ 1, ∈t T and ∈h H (7)
( ) i t pd 1
it h ith
iI t t
x xM
∈ = +1 ′
∑ ∑ ≤ 1 , ∈t T , i t pd T + ≤ and ∈h H (8)
3
ith
iGtT
x U ∈ ∈
∑∑ = ,∈h H (9)
∑ ≤ 1 ∈t T , t tg T + ≤ and ∈h H (10)
( )
i t pd
ith
iFp tt
∈ =
∑ ∑ ≤ 1, , i ∈ ∈ pNtS , i t pd T + ≤ and ∈h H (11)
' ,'
( )
ith ith h h
iFp
xx e ∈
∑ + ≤2, , i ∈ ∈ pNtS , ∈ hh H , ' , h h ≠ '
(12)
xith ∈ 01 { , } ,∈ ∈ iI tT , and ∈h H (13)
ijth y ≥ 0and ith z ≥ 0, , ,i ∈ ∈ ∈ iC tS jJ and ∈h H (14)
The objective function is to maximum the profit. Constraint (1) ensures that total
production costs are no more than the budgets. Constraint (2) defines the yield of vegetable i
planted in period t for unit area of plot h . Constraint (3) enables that the yield of vegetable i
in harvest period + i t pd ?1is zero if it is not planted in period t on plot h . Constraint (4)
ensures that yield of vegetable i must satisfy its demand. Constraint (5) guarantees that the total
quantity of fertilizer for vegetable i in production duration is zero if it is not planted in period
t on plot h . Constraint (6) ensures that the quantity of chemical input j is under the maximum
admissible level. Constraint (7) ensures that vegetable i ,? ∈i I cannot occupy the land at the
same time in each rotation. That is, at most one crop is planted at each period and each plot.
Constraint (8) guarantees that if a crop is planted, no other crop (including green manure) can
be planted during the time interval of production until the crop being harvested. Both constraint
(9) and Constraint (10) enforce that one green manure must be planted in each rotation.
Constraint (11) forbids that vegetables from the same botanic family are cultivated immediately
one after the other, while vegetables from the different botanic family, green manures and
fallow period are allowed. Constrain (12) ensures that vegetables from the same botanic family
are not cultivated on the adjacent plots. Constrain (13) and (14) enforce the restrictions on
decision variables.
因为专业，所以值得信赖。如有需要，请加QQ：99515681 或邮箱：[email protected]
微信：codinghelp