Optimization of cultivation pattern for
maximizing farmers' profits under land- and
water constraints by means of linearprogramming: An Iranian case study
Mohammad Zare
Manfred Koch
Dept. of Geotechnology and Geohydraulics,
University of Kassel, Kassel, Germany
Contents

Introduction

Literature review

Study area

Materials and methods

Results and discussion

Conclusions
1
Introduction



Agricultural development plans are usually faced with
the problem of how to find optimal cultivation pattern
for maximizing the economical profit of the farmers.
The simplest way would be to use some trial-and-error
method.
complicated optimization problem, as the one discussed
here, such a trial-and-error approach is neither practical
nor may it be able to find the optimal solution.
2
Introduction


Mathematical programming (MP) comprises a set of
techniques for dealing with specific constrained
optimization problems, as they arise in many branches
of management science.
When the mathematical representation of the
optimization problem can be cast in a linear form, the
general MP-model becomes a linear programming
(LP) model.
3
Introduction


LP is basically applied when the optimum allocation of
limited resources among competing activities, under a
set of linear constraints imposed by the nature of the
problem.
In the present paper the LP-method has been used for
the optimization of cultivation pattern in the 100 ha
irrigated farm of the Agriculture Faculty of Kermanshah
University, Iran.
4
Literature Review
Year
Researcher
Description
1947
Dantzig
developed the Simplex method for solving the
general linear-programming problem
Singh et al
formulated a LP-model for finding an optimal
cropping pattern, giving the maximum net return
at different water levels in a plain in India
2001
A LP-model was applied to farm data collected
Igwe and
from thirty crop farmers of the Aba Agricultural
2013
Onyenweaku Zone in the the Abia State, Nigeria, during the
2010 farming season.
5
Study area


The study area is located in Kermanshah City in western Iran
on the boarder of the Gharasu River. This region is
geographically limited in the North by the Faculty of
Agricultural sciences of Kermanshah University, in the
South and West by Kermanshah City and in East by the
Gharasu River. It has a surface area of about 100 ha.
The region is a semi-arid area, so the cultivation is entwined
with irrigation. The farm is irrigated by 7 wells that are deep
and semi deep. The irrigation time period is started from
April to October.
6
Materials and methods
Linear program (LP) modeling



Linear programming (LP) comprises a powerful set of tools
for linear constrained optimization.
LP is formulated by (1) an objective function and, (2) a set of
constraints (typically indicating resources limitations)
In the present study, because of the different spatial locations
of the 7 groundwater extraction wells, each of them supplying
only the farm area in its vicinity, LP- models for each of these
well- areas have been set up in the WINQSB environment.
Finally, the 7 individual LP-solutions will be accumulated to
get the optimal cultivation pattern for the entire farm plot
7
Materials and methods
Objective function

To set up the objective function Z the area Ai for each
crop must be multiplied by the net profit Pi per ha of
each crop which, in return, is calculated from the
difference of gross income and costs.
8
Max Z   Ai Pi
i 1
8
Materials and methods
Objective function

No
1
2
3
4
5
6
7
8
9

Cost items for crop cultivation
Item
Farm rent
Soil analyzing
Herbicides
Pesticide
Fungicides
Potash F*
Phosphorus F*
Nitrogen F*
Micronutrient F*
Unit
Ha
Item
L
L
Kg
Kg
Kg
Kg
L
No
10
11
12
13
14
15
16
17
18
Item
Transportation
Plow
Disk
Leveler
Lining
Seed
Seeding
Irrigation
weeding
Unit
Item
Item
Item
Item
Item
Kg
Item
Item
Item
No
19
20
21
22
23
24
25
26
27
Item
F application
Spraying
Cultivator
Harvester machine
haulm cutting
Loading
Insurance
Worker
Water price
Unit
Item
L
Item
Hour
Item
Ton/ha
€/ha
Person-hour
€/m3
Net profit Pi of each of the eight crops used in the LPmodel
Crop
Net Profit(€/ha)
Wheat
1060.9
Barley
646.37
Maize
737
Sunflower
-13.36
Soybean
213.83
Alfalfa
523.07
Canola
237.89
Sorghum
449.35
9
Materials and methods
Resource restrictions/constraints

The first constraint is related to the soil area AWj
covered by each of the j=1,..,7 groundwater wells.
8
A
i 1
ji
AW j
Well No.
Covered area (ha)
( j  1,...,7)
1
5
2
25
3
15
4
22
5
10
6
10
7
15
Sum
102
10
Materials and methods
Resource restrictions/constraints

The water availability is another constraint.
8
Crop/Month
Wheat
Barley
Maize
Sunflower
Soybean
Alfalfa
Canola
Sorghum
April
A NIR May
VW

194.19
323.34
i 1
96.74
0
0
0
41.65
30.52
0
ji
i
149.85
22.64
5.28
7.11
142.9
54.45
19.6
j
 eJune
355.92
128.14
128.87
26.21
35.12
259.66
97.42
108.3
July
50.77
0
167.99
42.87
46.05
304.63
103.2
140.57
August
0
0
109.83
37.38
43.99
304.29
0
84.34
September
0
0
0
4.55
13.56
223.06
0
0
October
41.58
20.71
0
0
0
123.92
19.51
0
where VWj denotes volume of groundwater withdrawal
from each well (m3/month)
e is the named surface irrigation efficiency that is equal
to 30%
NIRi is the water requirement of crop i for a particular
month (mm/month)
11
Materials and methods
Resource restrictions/constraints
The third constraint is related to the minimal farm
area of a specifically required crop that should be
considered in cultivation pattern. The farm belongs to
the Agricultural faculty of Kermanshah University
and which manages aviculture and livestock. As the
latter are fed by barely, alfalfa, sorghum and canola,
Crop
Alfalfa Barley Sorghum Canola
minimal cultivation areas for these crops must be
Minimum cultivated area (ha)
15
4
10
1.5
provided

12
Results and discussion
Optimum cultivation pattern

The LP- models for the individual well areas are solved
in the WINQSB environment and the optimal cultivation
pattern for these areas has been calculated
Well/Crop
Wheat
Barley
Maize
1
2
3
4
5
6
7
Sum
3.5
9.6
5
7.5
2.5
3.1
4.4
35.5
0
3
0
6.2
4.3
3.1
5.6
22.3
0
12.4
0
3.3
0.2
1.8
0
17.6
Sunflower
& Soybeans
0
0
0
0
0
0
0
0
Alfalfa
Canola
Sorghum
Sum
0
0
0
5
3
2
5
15
1.5
0
0
0
0
0
0
1.5
0
0
10
0
0
0
0
10
5
25
15
22
10
10
15
102
13
Results and discussion
Optimum cultivation pattern
14
Results and discussion
Net profits

with the optimal cultivation pattern, the net profit will be
increased by about 8000 Euro which, when compared with
the existing profits, amounts to an increase of 11.3%.
Profit/Crop
Net profit (€/ha)
Net profit/ OP (€)
Net profit/ EP (€)
Net profit diff. (€)
Wheat
1060.9
37662
31827
5835
Barley
646.4
14414
9695.5
4718.5
Maize
737
12971.3
14740.1
-1768.8
Sunflower
-133.6
0
-534.5
534.5
Soybean
213.8
0
855.3
-855.3
Alfalfa
523.1
7846
7846
0
Canola
237.9
356.8
951.5
-594.7
Sorghum
449.4
4493.5
4493.5
0
Sum
--77743.5
69874.5
7869
15
Results and discussion
Sensitivity analysis


Well/Mon
1
2
3
4
5
6
7
Sum
The sensitivity analysis on RHS of water constraints is
carried out in the WINQSB environment.
the optimal cultivation pattern allows for an additional
reduction of 52878 m3 water per year, without any
April
May
June of the
Julynet profit.
August
September
October
Sum
significant
decrease
1164
2566
3754
934
934
982
1613
11947
1648
3315
6209
1084
1084
1186
1897
16423
768
0
2537
0
0
0
100
3405
693
0
905
0
0
0
0
1598
1080
1340
1856
193
193
317
314
5293
972
2430
2430
341
341
564
537
7615
743
1832
2087
389
389
514
643
6597
7068
11483
19778
2941
2941
3563
5104
52878
16
Conclusions

The results of the LP- constrained maximization indicate that
the optimal cultivation areas for wheat, barley, maize,
alfalfa, sorghum and canola are 35.5, 22.3, 17.6 15, 10 and
1.5 Hectares, respectively, whereas those for soybean and
sunflower are essentially zero, which means that the
cultivation of these two crops is not economical and should
so be eliminated from future farm cultivation.
17
Conclusions


With optimal cultivation pattern, an 11.3% annual increase
of the economic profit can be gained, when compared with
that of the present cultivation scheme.
The results of sensitivity exercise indicate that -even with
this 11.3 % increased net income- with the optimum
cultivation pattern a further reduction of 52878m3 of water
(equal to 11.9 percent of the total available water) can be
achieved per year. This amount of water could supply a
portion of the domestic water needs of the Faculty of
Agriculture of the University.
18
THANK YOU
FOR YOUR
ATTENTION