researcharticle linear optimization model for efficient...

Research ArticleLinear Optimization Model for Efficient Use of Irrigation Water

Wafa Difallah1 Khelifa Benahmed2 Belkacem Draoui1 and Fateh Bounaama1

1Laboratory of Energetic in Arid Zones (ENERGARID) Department of Electrical Engineering Faculty of TechnologyTahri Mohammed University of Bechar Bechar Algeria2Department ofMathematics andComputer Science Faculty of Exact Sciences TahriMohammedUniversity of Bechar Bechar Algeria

Correspondence should be addressed to Wafa Difallah wafadifgmailcom

Received 8 May 2017 Accepted 27 June 2017 Published 26 July 2017

Academic Editor David Clay

Copyright copy 2017 Wafa Difallah et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The implementation of innovative and efficient irrigation techniques is among the greatest challenges facing agriculture In thisregard a linear programming model is presented in order to optimize water use The idea behind this model is to assess theeffectiveness or ineffectiveness of precipitation to determine the amount of irrigation water required to optimize water use Toachieve this idea the ldquoknapsackrdquo problem decisional form was used and the combination of the linear programming and theabove-mentioned form proved satisfactory Field experiments were conducted in Algeria Based on calculated budgets a modelusing linear programming was developed A comparison between the model results and the field findings suggests that the modelcould reduce water consumption by 285

1 Introduction

Irrigation is the most used means of agricultural intensifi-cation and will stay a cornerstone in the domain of foodsecurity policies towards climatic variability [1] It can bedefined as artificial implementation of water in agricultureand is regarded as a very significant constituent of agrarianactivity [2]

In most countries water is used for irrigating land morethan for any other purpose and any decrease in water supplycan cause production and yield to decrease

In order to reduce water consumption without compro-mising agricultural yield as an economic mainstay variousstudies have been carried out by researchers who haddifferent visions on the subject

Some researchers attempted to estimate evapotranspira-tion to determine water needs and hence to optimize theuse of irrigation water A neural network approach wasproposed by Laaboudi et al [3] to estimate the referenceevapotranspiration and resolve the problem of the high num-ber of climatic parameters needed to use Penman Montheithequation Takakura et al [4] used the energy balance equationto measure evapotranspiration Takakura et al proved intheir study the simplicity and suitability of their method forirrigation

For the same purpose support vector machine methodwas used by Yao et al [5] and artificial neural networkstechnique was used by King et al [6] Extraterrestrial radi-ation air temperature and humidity were used in [7] toestimate potential evapotranspiration and to assess futureclimate change effects on the vegetation

Other researchers insist on the necessity of consideringthe economic factors [8 9] to decide about irrigationdepths and select crop zones for economic benefit Kuo ampLiu [8] developed a customized genetic algorithm For thesame purpose the exact optimization methods were alsoused to mention but a few linear programming [10ndash12]nonlinear programming [13] quadratic programming [14]and dynamic programming [9]

Many authors used real time monitoring applications[15] by means of wireless sensor networks (WSNs) Thesenetworks make possible a timely and minute irrigationapplication according to the existing environmental condi-tions resource availability and weather forecasts [16 17]The potential applications of WSN in irrigation cover alarge set of scenarios and applications [18] For instance areal time intelligent irrigation controller has been conceivedin order to detect the optimal irrigation time [19] Forthe same purpose a wireless data acquisition network was

HindawiInternational Journal of AgronomyVolume 2017 Article ID 5353648 8 pageshttpsdoiorg10115520175353648

2 International Journal of Agronomy

implemented and applied to control drip irrigation of dwarfcherry trees [20] Angelopoulos et al [21] minimized thestudy area and proposed a smart home-irrigation systemThis system is managed to maintain soil humidity levels andcover the different watering needs Ant Colony Algorithmwas also combined with a wireless sensor network to identifyareas which are relatively arid and avoid water wastage [22]Another implementation of WSN for two kinds of soil andsix different crops was performed by Kumari et al [23] Gaoet al [24] combined WSN with a fuzzy control system in anintelligent water-saving irrigation system in their research toachieve remote on-line monitoring and control

To highlight the impact of WSN use in irrigation acomparison study between automated drip irrigation sys-tem and non-automated drip irrigation systems was doneby Soorya et al [25] This study proved that the auto-mated system exhibited more promising results comparedto the other system However WSNs consist of battery-powered nodes known for limited energy capacity [26] asa result several studies about solar powered nodes in irri-gation were proposed as a solution to the energy constraint[20 27]

Among the various approaches presented the linearprogramming model is considered as the most heavily usedtechnology in resource management such as the problemposed in this study Therefore this model has been selectedin order to ensure an integrated irrigation management

Thenew feature in this proposal is to take advantage of theknapsack optimization problem which dates back to morethan a hundred years to solve a contemporary problemwhichis water shortage The object of this proposal is to develop anapproach which uses the right amount of water and replacethe traditional strategies of regular irrigation whether theplants need it or not

Besides the discussion of previous works performedin water optimization this study comprises five sectionsthe first presents factors that affect plant water needs thesecond deals with the model description followed by modelformulation using linear programming next a numericalstudy will be done using Matlab R2015a Software to validatethe performance of the proposed model and the last sectionconcludes the work

2 Materials and Methods

21 Experimental Setting The study of cereals was conductedin the meteorological station of Ain Skhouna municipalityof Zarma was selected it is located 23Km east of the cityof Batna in Algeria The variety of the sample crop (winterwheat) used was sown on November 20th and harvested onJune 10th The climate is semiarid with an annual averagerainfall of 327mm insufficient for the cereals due to a dryperiod of 06 months from May to November and thusrequiring irrigation (see Figure 1)The soil is not saline and isrich in limestone with a slightly basic pH (71) in a fine clay-loam texture [28]

22 Determining Plant Irrigation Needs Determining thetiming and amount of necessary water for irrigation (plant

0

10

20

30

40

50

Rain

fall

(mm

)

MayApJan JunFD MNMonths

Figure 1 Monthly changes of rainfall in the study field

irrigation needs) can be a complex process that takes intoconsideration factors such as the following

221 Climatic Conditions Prevailing in the Region The effectof weather on plant water needs is given by the referencecrop evapotranspiration ldquoET0rdquo which can be defined asthe rate of evapotranspiration from a wide surface of 8 to15 cm tall green grass cover of identical height activelygrowing completely shading the ground and not short ofwater It is expressed in mm per day [29] ET0 is calculatedusing (1) which takes into account the climatic parametersof temperature solar radiation wind speed and humidityThis equation was estimated from the Penman-Montheithformula given in (2)

ET0

=0408Δ (119877119899 minus 119866) + 120574 (900 (119879 + 273)) 1199062 (119890119904 minus 119890119886)

Δ + 120574 (1 + 0341199062)

(1)

120582ET =Δ (119877119899 minus 119866) + 120588119886119888119901 ((119890119904 minus 119890119886) 119903119886)

Δ + 120574 (1 + 119903119904119903119886) (2)

where ET0 is the reference evapotranspiration [mmdayminus1]119877119899 is the net radiation at the crop surface [MJmminus2 dayminus1] 119866is the soil heat flux density [MJmminus2 dayminus1] 119879 is the mean ofdaily air temperature at 2m height [∘C] 1199062 is the wind speedat 2m height [m sminus1] 119890119904 is the saturation vapour pressure[kPa] 119890119886 is the actual vapor pressure [kPa] 119890119904 minus 119890119886 is thesaturation vapor pressure deficit [kPa] Δ is the slope vaporpressure curve [kPa ∘Cminus1] 120574 is the psychometric constant[kPa ∘Cminus1] 120582ET is the latent heat flux [MJ kgminus1] 120588119886 is themean air density [gmminus3] 119888119901 is the specific heat of the air[MJ kgminus1 ∘Cminus1] 119903119886 is the aerodynamic resistance [smminus1] and119903119904 is the bulk surface resistance [sm

minus1] [30 31]

222 Plant Type and Stage of Growth Different plantsrequire specific amounts of water at various stages of theirgrowth and as mentioned in [32] the crop evapotranspira-tion ET119888 is calculated by multiplying the reference evapo-transpiration ET0 and the agricultural coefficient ldquo119896119888rdquo which

International Journal of Agronomy 3

EUR

Evaporation

Transpiration

Irrigation

Precipitation

Evapotranspiration

Drainage

Runoff

URMax

Figure 2 Water balance model

is different from one plant to another and from one growthphase to another

ET119888 = 119896119888 times ET0 (3)

223 The Properties of the Soil The type of soil plays amajor role in determining the plant water balance Thesoil water content is mainly due to its texture which isdetermined uniquely by a combination of three variablessand percentage silt and clay content of the soil [33]

Water retention capacity depends on the composition andcompaction of the soil Crops sown in sandy soils need morewater because of higher infiltration rate Likewise surfacesoils need more water compared to deeper soils [34]

224 Irrigation Method To select a convenient irrigationstrategy it is necessary to be aware of the benefits andshortcomings of these strategies The agriculturer has to beable to decide about the best strategy to use according to localconditionsHowever it is often difficult to find the best optionsince any strategy has benefits and shortcomings [35]

The choice of the irrigation strategy that is using surfacesprinkler or drip irrigation is mainly based on naturalconditions the kind of plant the available techniques priorexperience the available labour force and the cost function[36]

23 Model Description There are different sources of waterto meet the needs of plants (see Figure 2)

First rain which is a natural source significantly affectingthe amount of water to be added to the plant is efficientwhen rainfall exceeds a threshold 1198791 and is less than 1198792

Table 1 Thresholds value

1198791 119890119894 minus 119904119894 minus 119894119894 + 1198891198941198792 119906119903119894 minus 119904119894 minus 119894119894 + 1198891198941198793 1198791 le 1198793 le 11987921198794 119890119894 minus 119903119894 minus 119894119894 + 1198891198941198795 119906119903119894 minus 119903119894 minus 119894119894 + 1198891198941198796 119890119894 minus 119903119894 minus 119904119894 + 1198891198941198797 119906119903119894 minus 119903119894 minus 119894119894 + 119889119894

or when the amount of rainfall of three consecutive daysexceeds a threshold1198793 (see Table 1) If not the rain is qualifiedas isolated and therefore not useful for the plant Thesethreshold values vary depending on the climate and soil ofthe agricultural zone for example any isolated rain less than10mm or any episode where the amount of rain over threedays is less than 15mm Similarly the excessive volume of rainstored out of the reach of the roots cannot be considered [37]

Another source is the soil reserve which plant roots canreach

Since soil oxygen is essential for root function the growthof plants depends on the H2O content and oxygenmovementwithin the soil [38]

Water in the soil can be classified as available or unavail-able water based on the amount held in the soil betweenretention point and wilting point water volume can varyfrom a small percentage to more than 50 percent [39] Theamount of available water depends on the soil textures andthe rooting system of the crop

According to Tiercelin and Vidal [37] the useful watercontent 120579119906 is given by

120579119906 = 120579119888119903 minus 120579119901119891 (4)

where 120579119906 is useful water content and 120579119888119903 is Retention pointthat is water content in the retention capacity If 120579119906 equals120579119888119903 there is an excess of water 120579119901119891 is wilting point or watercontent capacity of wilting If 120579119906 equals 120579119901119891 there is a lack ofwater

It is also possible to calculate 120579119906 using the equation ofuseful water reserve

UR = (120579119888119903 minus 120579119901119891) times 119885

120579119906 = UR119885

(5)

where UR is useful water reserve and 119885 is depth of soilreached by the roots

And themost important resource is thewater provided bythe agriculturer which is generally a limited resource whichmust be used with precaution

This resource represents the difference between the totalamount of water needed by the plant and the water providedby rainfall and the soil

In addition to the discussed resources plant waterneeds include all losses such as evapotranspiration whichis expressed by (1) and losses during irrigation expressedin terms of drainage water and represented by ldquo119863rdquo in theproposed model


Start

End

YesYes

Yes

ii = ni minus ri minus si + di

T1 le ri le T2

ri + minus di ge ni

ii = ni minus si + di

si minus di ge ni

Drainage (di)i)

Soil moisture (si)Get the controlled values

NoNo

No

Rainfall (ri)Needed water (n

si

ii = 0

Figure 3 Flow chart of the proposed model

The design of the model suggested in this work is gener-ally based on the flowchart illustrated in Figure 3The idea ofthe system is to take advantage of all available water resourcesto determine the optimal quantity of irrigation water and asnoted in Figure 3 the system handles all conditions that maybe faced by the agriculturer during the irrigation process

Case 1 When there is no rainfall and the soil moistureis insufficient to cover the lost water the irrigation waterprovided by the agriculturer will be equal to the differencebetween plant water requirement and soil water content

Case 2 If the soil moisture is equivalent to the plant waterneeds either with or without rainfall in this case theirrigation water will take the value 0

Case 3 If rain and soil moisture are not effective and thereis a lack of coverage of the plant water needs the value ofirrigationwater will be equal to the difference needed to coverthis lack

An analysis of the efficiency or inefficiency of precipita-tion and soil moisture will be discussed in the next section

24 Model Formulation In this study a linear programmingmodel is used A set of techniques and methods is inferredfrom mathematics and other sciences which was developedduring World War II and it is continually applied byresearchers to improve management decisions [40]

The formulation process model contains

(i) the identification of the decision variables(ii) the formulation of the objective function

(iii) the identification and the formulation of the con-straints

Let 119875 = [119905 119905 + 1 119905 + 2 119879] be planting period and itvaries from one plant to another and can be measured withany usual temporal unit in which ldquo119905rdquo is the planting timeand ldquo119879rdquo is the harvest time NW = [119899119894 119899119894+1 119899119894+2 119899119879]is needed water and it refers to the water lost throughevapotranspiration 119877 = [119903119894 119903119894+1 119903119894+2 119903119879] is efficient rainURMax = [119906119903119894 119906119903119894+1 119906119903119894+2 119906119903119879] is maximal useful waterreserve EUR = [119890119894 119890119894+1 119890119894+2 119890119879] is easily used waterreserve 119878 = [119904119894 119904119894+1 119904119894+2 119904119879] is soil moisture and 119868 =[119894119894 119894119894+1 119894119894+2 119894119879] is irrigation water These values are thedecision variables of the linear optimization model 119863 =[119889119894 119889119894+1 119889119894+2 119889119879] is drainage water in which

120575119899120575119905

= 120575119903120575119905

+ 120575119904120575119905

+ 120575119894120575119905

minus 120575119889120575119905 (6)

241 Identification of the Objective Function The aim of theproposedmodel is tominimize the quantity of water requiredfor irrigation using soil humidity and rain as explainedin

Min119879

sum119894=119905

1205721119903119894 + 1205722119904119894 + 119894119894 (7)

1205721 = 1 for 119903119894 isin 119877 and 1205722 =1 for 119906119903119894 isin UR

242 Identification and Formulation of the Constraints

(1) Limits of Efficient Rainfall

119903119894 isin 119877 997888rarr 119903119894 ge 0

1198791 le 119903119894 le 1198792 forall119894 isin [119905 119879]

119894+2

sum119905=119894

119903119894 ge 1198793 forall119894 isin [119905 119879 minus 2]

(8)

(This condition applies just when there is successive rainfall)

(2) Limits of Useful Reserve

119906119903119894 isin URMax 997888rarr 119906119903119894 ge 0

120579119901119891 lt 119906119903119894 lt 120579119888119903 forall119894 isin [119905 119879] (9)

The same conditions remain valid for the EUR given thatthe latter is less than URMax

(3) Limits of Soil Humidity

119904119894 isin 119904 997888rarr 119904119894 ge 0

1198794 le 119904119894 le 1198795 forall119894 isin [119905 119879]

119904119879+1 = 0

(10)


Table 2 Summary of monthly results during the development cycle (JanuaryndashJune)

Month Jan Feb Mar Apr May JunRooting depth (cm) 12ndash17 18ndash215 22ndash35 36ndash455 46ndash53 53EUR (mm) 4287 5517 7250 11250 14133 5017119877 (mm) 224 122 145 468 60 32119878 (mm) 2143 2758 3625 5625 7067 2508119863 (mm) 4525 4625 355 10225 850 420URMax (mm) 643 8275 10875 16875 2120 7525NW (mmmonth) 421 4507 915 11392 13325 3325

Table 3 Usual and estimated irrigation water from January to June (mm)

Month Jan Feb Mar Apr May Jun TotalUsual irrigation water 7525 212 16875 10875 8275 6430 71180Estimated irrigation water 6389 14966 11312 7625 6164 4429 50885

(4) Limits of Available Irrigation Water

119879

sum119894=119905

119894119894 le 119868

119894119894 isin 119868 997888rarr 119894119894 ge 0

1198796 le 119894119894 le 1198797

(11)

Threshold values vary from one agricultural zone toanother according to climatic conditions These values arecomputed as illustrated in Table 1

Therefore the problem was reduced to the followinglinear program

Minimize Z =119879

sum119894=119905

1205721119903119894 + 1205722119904119894 + 119909119894 (12)

Subject to119879

sum119894=119905

119909119894 le 119868 (13)

119909119894 ge 119890119894minus119903119894 minus 119904119894 + 119889119894 (14)

119909119894 le 119906119903119894minus119903119894 minus 119904119894 + 119889119894 (15)

119903119894 ge 119890119894minus119909119894 minus 119904119894 + 119889119894 (16)

119903119894 le 119906119903119894minus119909119894 minus 119904119894 + 119889119894 (17)

119904119894 ge 119890119894minus119903119894 minus 119909119894 + 119889119894 (18)

119904119894 le 119906119903119894minus119903119894 minus 119909119894 + 119889119894 (19)

119903119905 + 119904119894 + 119909119894 minus 119889119894 ge 119899119894 (20)

119904119879+1 = 0 (21)

1205721 1205722 isin [0 1] (22)

119909119894 119890119894 119903119894 119904119894 119906119903119894 119889119894 119899119894 ge 0

forall119894 isin [119905 119879] (23)

In this model

(i) the objective function (12) with constraints (13) (14)(15) and (20) minimizes the quantity of irrigationwater

(ii) constraints (16) and (17) are used to check rainfallefficiency

(iii) constraints (18) and (19) are used to check the effi-ciency of the soil reserve in water (soil moisture)

(iv) constraint (21) is used to ensure that at the end of theplanting season the soil reserve is fully used thus it iswell exploited in order to optimize irrigation water

3 Results

In order to highlight the necessity of using a linear program-ming model for irrigation water optimization it is necessaryto first determine the values of each term of the proposedmodel as described in Section 24

Results in Table 2 illustrate the monthly variables overthe experimental study field used to resolve the optimizationproblem

In order to check the validity of the proposed model theobtained results will be compared with the usual amount ofwater used for irrigation during the experimental periodThese values are based on traditional methodsThe usual andestimated irrigation water distribution from January to Juneis presented in Table 3

4 Discussion

In the presented proposal a linear programming model wascombined with the ldquoKnap sackrdquo problem decisional form toevaluate water use efficiency by studying the effectiveness orineffectiveness of rainfall and soil water content

To resolve the optimization problem the linear pro-gramming function of MATLAB R2015a software is usedThe obtained results indicate that the total irrigation watermeasured using the proposed model is less than the totalusual quantity with 20295mm and this is also confirmed


Usual I (mm)Estimated I (mm)

0

50

100

150

200

250

MayAp JunJan F MMonths

Figure 4 Usual and estimated irrigation water

10

20

30

40

50

60

70

Irrig

atio

n w

ater

(mm

)


Figure 5 Monthly difference of estimated and usual irrigationwater

in Figure 4 which shows the difference between usualand estimated irrigation water during different plantationperiods and this has to do with the full use of all availablewater resources either in the soil or as rainfall as describedin the system constraints

Figure 5 shows the monthly difference of estimated andusual irrigation water from January to June It can clearlybe seen that there has been a positive difference betweenusual and estimated irrigationwaterThis difference increasesto reach the highest value in the month of May which isestimated at 6234mm and then decreases until the monthof April to 1136mm and this is a logical decrease becauserainfall in this period reaches the lowest levels as shown inFigure 1

According to the system constraints the lowering in thevalue of 119894119894 means more economy in water as a source ofirrigation Moreover the increase of this value which is dueto inadequate 119903119894 and 119904119894 to cover the plantrsquos needs means thatthere is a renewal of soil water which helps prevent thephenomenon of underground water pollution

Moreover by looking at Figure 6 it can be noted thatthere is a direct proportion between rainfall and the dif-ference between the losses due to the evapotranspirationand drainage on the one hand and the estimated irrigation

MonthsJan F M Ap May Jun

0

20

40

60

80

100

120

Rainfall (mm)ETc + D minus estimated I (mm)

Figure 6 Relation between (ET119888 + 119863 minus estimated 119868) and Rainfall

water on the other which means that there is a decrease inthe amount of irrigation water accompanied with increasein rainfall which proves that there is an optimized useof rain water as an alternative resource for irrigation andthis is one of the most important features of the proposedmodel

The results of this study have revealed the importance ofirrigation water management especially in arid and semi-arid regions under extreme climatic conditions The studytakes advantage of the positive impact of precipitation andsoil water content to improve the efficiency of water use

5 Conclusion

The present paper attempted to develop a linear program-ming model that improves water use efficiency based onweather and soil conditions

The model in its structure is based on the use of simpleequations which makes it attractive to use by a wide varietyof agriculturers

As shown in this study the model can be used for agri-cultural purposes and the experimental results demonstratethat by applying the proposed optimization model it ispossible to reduce water consumption by 2851

Furthermore excessive irrigation can lead to groundwater pollution and to salinization of this resource Hencethis method can reduce these effects through rational use ofwater

In addition to that this model is feasible to apply inany area the difference lies only in the threshold valuescharacterizing each zone

It is worth mentioning that this system should be furtherdeveloped so that it will have more features and can be com-bined with recent technological advances like the Internet ofthings in order to monitor and schedule irrigation water

Moreover this system could be developed for use insidegreenhouses where it is possible to control external factorssuch as relative humidity temperature wind speed andsunshine duration and also in order to enrich the system


with other parameters that might be the subject of furtherinvestigation

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] FAO water reports Climate change water and food securityFood and Agriculture Organization of the United NationsRome Italy 2011

[2] Aqeel-ur-Rehman et al ldquoA review of wireless sensors andnetworksrsquo applications in agriculturerdquo Computer StandardsInterfaces vol 36 pp 263ndash270 2014

[3] A Laaboudi B Mouhouche and B Draoui ldquoNeural networkapproach to reference evapotranspiration modeling from lim-ited climatic data in arid regionsrdquo International Journal ofBiometeorology vol 56 no 5 pp 831ndash841 2012

[4] T Takakura C Kubota S Sase et al ldquoMeasurement of evapo-transpiration rate in a single-span greenhouse using the energy-balance equationrdquo Biosystems Engineering vol 102 no 3 pp298ndash304 2009

[5] Y Yao S Liang X Li et al ldquoImproving global terrestrialevapotranspiration estimation using support vectormachine byintegrating three process-based algorithmsrdquo Agricultural andForest Meteorology vol 242 pp 55ndash74 2017

[6] D A King D M Bachelet A J Symstad K Ferschweilerand M Hobbins ldquoEstimation of potential evapotranspirationfrom extraterrestrial radiation air temperature and humidityto assess future climate change effects on the vegetation of theNorthern Great Plains USArdquo Ecological Modelling vol 297 pp86ndash97 2015

[7] V Z Antonopoulos and A V Antonopoulos ldquoDaily refer-ence evapotranspiration estimates by artificial neural networkstechnique and empirical equations using limited input climatevariablesrdquoComputers and Electronics in Agriculture vol 132 pp86ndash96 2017

[8] S-F Kuo and C-W Liu ldquoSimulation and optimization modelfor irrigation planning and managementrdquo Hydrological Pro-cesses vol 17 no 15 pp 3141ndash3159 2003

[9] Z X Fang B Voron and C Bocquillon ldquoDynamic program-ming a model for an irrigation reservoirrdquoHydrological SciencesJournal vol 34 no 4 pp 415ndash424 1989

[10] J Auriol et al ldquoOptimization of catchment devices for simu-lation irrigation on mathematical models and linear program-mingrdquo in Proceedings of the Paper presented at the 9th WorldCongress of the ICI Moscow 1975

[11] Z Zhenmin ldquoOptimization of water allocation in canal systemsof ChenGai irrigation areardquo inProceedings of the Paper presentedat CEMAGREF-IIMI InternationalWorkshop onTheApplicationof Mathematical Modeling for the Improvement of Irrigationcanal Operation Montpellier France 1992

[12] A Singh and S N Panda ldquoOptimization and SimulationModelling for Managing the Problems of Water ResourcesrdquoWater ResourcesManagement vol 27 no 9 pp 3421ndash3431 2013

[13] V Azimi et al ldquoOptimization of deficit irrigation using non-linear programming (Case studyMianeh region Iran)rdquo Inter-national journal of Agriculture and Crop Sciences vol 6 no 5pp 252ndash260 2013

[14] S Hong et al ldquoOptimization of irrigation scheduling forcomplex water distribution using mixed integer quadratic pro-gramming (MIQP)rdquo in Proceedings of the the 10th InternationalConference on Hydroinformatics Hamburg Germany 2012

[15] P Upadhyay et al ldquoEvaluating seed germination monitoringsystem by application of wireless sensor networks a surveyrdquoAdvances in Intelligent Systems and Computing vol 411 pp 259ndash266 2016

[16] K R Thorp D J Hunsaker A N French E Bautista and KF Bronson ldquoIntegrating geospatial data and cropping systemsimulation within a geographic information system to analyzespatial seed cotton yield water use and irrigation require-mentsrdquo Precision Agriculture vol 16 no 5 pp 532ndash557 2015

[17] D Levy W K Coleman and R E Veilleux ldquoAdaptation ofpotato to water shortage irrigation management and enhance-ment of tolerance to drought and salinityrdquo American Journal ofPotato Research vol 90 no 2 pp 186ndash206 2013

[18] J M Barcelo-Ordinas ldquoA survey of wireless sensor technologiesapplied to precision agriculturerdquo in Precision agriculture 13 J VStafford Ed Wageningen Academic Publishers 2013

[19] Dassanayake et al ldquoWater saving through smarter irrigation inAustralian dairy farming Use of intelligent irrigation controllerand wireless sensor networkrdquo in Proceedings of the Paperpresented at the 18thWorld IMACS MODSIMCongress CairnsAustralia 2009

[20] M Dursun and S Ozden ldquoA wireless application of dripirrigation automation supported by soil moisture sensorsrdquoScientific Research and Essays vol 6 no 7 pp 1573ndash1582 2011

[21] C M Angelopoulos S Nikoletseas and G CTheofanopoulosldquoA Smart system for garden watering using wireless sensor net-worksrdquo in Proceedings of the 9th ACM International Symposiumon Mobility Management and Wireless Access MobiWacrsquo11 Co-located with MSWiMrsquo11 pp 167ndash170 usa November 2011

[22] S Sutar et al ldquoIrrigation and fertilizer control for precisionagriculture using wsn energy efficient approachrdquo InternationalJournal of Advances in Computing and Information Researchesvol 1 no 1 2012

[23] R N Kumari et al ldquoMicrocontroller based irrigation usingsensor [a smart way for irrigation]rdquo International Journal ofResearch in Engineering ampamp Advanced Technology vol 2 no2 pp 1ndash3 2014

[24] LGaoMZhang andGChen ldquoAn intelligent irrigation systembased on wireless sensor network and fuzzy controlrdquo Journal ofNetworks vol 8 no 5 pp 1080ndash1087 2013

[25] E Soorya et al ldquoSmart drip irrigation system using sensornetworksrdquo International Journal of Scientific ampamp EngineeringResearch vol 4 no 2 pp 2039ndash2042 2013

[26] Y THou Y Shi andHD Sherali ldquoRate allocation and networklifetime problems for wireless sensor networksrdquo IEEEACMTransactions on Networking vol 16 no 2 pp 321ndash334 2008

[27] P Alagupandi R Ramesh and S Gayathri ldquoSmart irrigationsystem for outdoor environment using Tiny OSrdquo in Proceedingsof the 3rd IEEE International Conference on Computation ofPower Energy Information and Communication ICCPEIC 2014pp 104ndash108 ind April 2014

[28] R Bourkouche ldquoLa consommation en eau du ble waterconsumption of wheatrdquo Revue Agriculture pp 205ndash209 2016httprevue-agrouniv-setifdzdocumentsnumero-specialsession5bourkouchepdf

[29] M Beg ldquoPrediction of crop water requirement a ReviewrdquoInternational Journal of AdvancedTechnology in Engineering andScience vol 2 no 1 pp 727ndash734 2014


[30] A Laaboudi et al ldquoConceptual reference evapotranspirationmodels for different time stepsrdquo Pet EnvironBiotechnol vol 3no 4 pp 1ndash8 2012

[31] FAO document Crop evapotranspiration - Guidelines for com-puting crop water requirements Natural ResourcesManagementand Environment Department httpwwwfaoorgdocrepX0490Ex0490e08htm

[32] R G Alen et al albakhr nutah lilmahasil dalil taqdir alaihtiajatalmayiya ldquo Crop Evapotranspiration Guidelines for Comput-ing crop Water Requirement rdquotarjama FAawad w Malssueudalnnashr aleilmi w almutabie- jamieat almalik seud almamlakatalearabiat alssaeudia 2006

[33] B J Cosby G M Hornberger R B Glapp and T R Ginn ldquoAstatistical exploration of the relationships of soil moisture char-acteristics to the physical properties of soilsrdquo Water ResourcesResearch vol 20 no 6 pp 682ndash690 1984

[34] Raghav ldquo5 Important Factors Affecting the Demand ofWater for Cropsrdquo httpwwwgeographynotescomarticles5-important-factors-affecting-the-demand-of-water-for-crops650

[35] W L Powers andM P lewis ldquoIrrigation Requirement of ArableOregon Soils Oregon State Collegerdquo 1941 httpirlibraryor-egonstateeduxmluibitstreamhandle195715005StationBul-letin394pdf

[36] C Brouwer et al Irrigation Water Management IrrigationMethodsTraining manual no 5 FAO Land and Water Develop-ment Division 2001

[37] J R Tiercelin andA VidalTraite drsquoirrigation ldquoIrrigation treatyrdquo2nd edition 2006

[38] J T Ritchie ldquoSoil water balance and plant water stressrdquo inUnderstanding Options for Agricultural Production vol 7 ofSystems Approaches for Sustainable Agricultural Developmentpp 41ndash54 Springer Netherlands Dordrecht 1998

[39] C M K Gardner et al Soil physical constraints to plantgrowth and crop production FAO Land and water developmentdivision 1999

[40] M S Lord ldquoLinear Programming Optimizing the ResourcesrdquoInterdisciplinary Journal of Contemporary research in businessvol 4 pp 701ndash705 2013

Submit your manuscripts athttpswwwhindawicom

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implemented and applied to control drip irrigation of dwarfcherry trees [20] Angelopoulos et al [21] minimized thestudy area and proposed a smart home-irrigation systemThis system is managed to maintain soil humidity levels andcover the different watering needs Ant Colony Algorithmwas also combined with a wireless sensor network to identifyareas which are relatively arid and avoid water wastage [22]Another implementation of WSN for two kinds of soil andsix different crops was performed by Kumari et al [23] Gaoet al [24] combined WSN with a fuzzy control system in anintelligent water-saving irrigation system in their research toachieve remote on-line monitoring and control

To highlight the impact of WSN use in irrigation acomparison study between automated drip irrigation sys-tem and non-automated drip irrigation systems was doneby Soorya et al [25] This study proved that the auto-mated system exhibited more promising results comparedto the other system However WSNs consist of battery-powered nodes known for limited energy capacity [26] asa result several studies about solar powered nodes in irri-gation were proposed as a solution to the energy constraint[20 27]

Among the various approaches presented the linearprogramming model is considered as the most heavily usedtechnology in resource management such as the problemposed in this study Therefore this model has been selectedin order to ensure an integrated irrigation management

Thenew feature in this proposal is to take advantage of theknapsack optimization problem which dates back to morethan a hundred years to solve a contemporary problemwhichis water shortage The object of this proposal is to develop anapproach which uses the right amount of water and replacethe traditional strategies of regular irrigation whether theplants need it or not

Besides the discussion of previous works performedin water optimization this study comprises five sectionsthe first presents factors that affect plant water needs thesecond deals with the model description followed by modelformulation using linear programming next a numericalstudy will be done using Matlab R2015a Software to validatethe performance of the proposed model and the last sectionconcludes the work

2 Materials and Methods

21 Experimental Setting The study of cereals was conductedin the meteorological station of Ain Skhouna municipalityof Zarma was selected it is located 23Km east of the cityof Batna in Algeria The variety of the sample crop (winterwheat) used was sown on November 20th and harvested onJune 10th The climate is semiarid with an annual averagerainfall of 327mm insufficient for the cereals due to a dryperiod of 06 months from May to November and thusrequiring irrigation (see Figure 1)The soil is not saline and isrich in limestone with a slightly basic pH (71) in a fine clay-loam texture [28]

22 Determining Plant Irrigation Needs Determining thetiming and amount of necessary water for irrigation (plant

0

10

20

30

40

50

Rain

fall

(mm

)

MayApJan JunFD MNMonths

Figure 1 Monthly changes of rainfall in the study field

irrigation needs) can be a complex process that takes intoconsideration factors such as the following

221 Climatic Conditions Prevailing in the Region The effectof weather on plant water needs is given by the referencecrop evapotranspiration ldquoET0rdquo which can be defined asthe rate of evapotranspiration from a wide surface of 8 to15 cm tall green grass cover of identical height activelygrowing completely shading the ground and not short ofwater It is expressed in mm per day [29] ET0 is calculatedusing (1) which takes into account the climatic parametersof temperature solar radiation wind speed and humidityThis equation was estimated from the Penman-Montheithformula given in (2)

ET0

=0408Δ (119877119899 minus 119866) + 120574 (900 (119879 + 273)) 1199062 (119890119904 minus 119890119886)

Δ + 120574 (1 + 0341199062)

(1)

120582ET =Δ (119877119899 minus 119866) + 120588119886119888119901 ((119890119904 minus 119890119886) 119903119886)

Δ + 120574 (1 + 119903119904119903119886) (2)

where ET0 is the reference evapotranspiration [mmdayminus1]119877119899 is the net radiation at the crop surface [MJmminus2 dayminus1] 119866is the soil heat flux density [MJmminus2 dayminus1] 119879 is the mean ofdaily air temperature at 2m height [∘C] 1199062 is the wind speedat 2m height [m sminus1] 119890119904 is the saturation vapour pressure[kPa] 119890119886 is the actual vapor pressure [kPa] 119890119904 minus 119890119886 is thesaturation vapor pressure deficit [kPa] Δ is the slope vaporpressure curve [kPa ∘Cminus1] 120574 is the psychometric constant[kPa ∘Cminus1] 120582ET is the latent heat flux [MJ kgminus1] 120588119886 is themean air density [gmminus3] 119888119901 is the specific heat of the air[MJ kgminus1 ∘Cminus1] 119903119886 is the aerodynamic resistance [smminus1] and119903119904 is the bulk surface resistance [sm

minus1] [30 31]

222 Plant Type and Stage of Growth Different plantsrequire specific amounts of water at various stages of theirgrowth and as mentioned in [32] the crop evapotranspira-tion ET119888 is calculated by multiplying the reference evapo-transpiration ET0 and the agricultural coefficient ldquo119896119888rdquo which


EUR

Evaporation

Transpiration

Irrigation

Precipitation

Evapotranspiration

Drainage

Runoff

URMax



ET119888 = 119896119888 times ET0 (3)














120579119906 = 120579119888119903 minus 120579119901119891 (4)



UR = (120579119888119903 minus 120579119901119891) times 119885

120579119906 = UR119885

(5)






Start

End

YesYes

Yes


T1 le ri le T2

ri + minus di ge ni


si minus di ge ni

Drainage (di)i)


NoNo

No


si

ii = 0












120575119899120575119905

= 120575119903120575119905

+ 120575119904120575119905

+ 120575119894120575119905

minus 120575119889120575119905 (6)


Min119879

sum119894=119905

1205721119903119894 + 1205722119904119894 + 119894119894 (7)




119903119894 isin 119877 997888rarr 119903119894 ge 0

1198791 le 119903119894 le 1198792 forall119894 isin [119905 119879]

119894+2

sum119905=119894


(8)




120579119901119891 lt 119906119903119894 lt 120579119888119903 forall119894 isin [119905 119879] (9)



119904119894 isin 119904 997888rarr 119904119894 ge 0

1198794 le 119904119894 le 1198795 forall119894 isin [119905 119879]

119904119879+1 = 0

(10)







119879

sum119894=119905

119894119894 le 119868

119894119894 isin 119868 997888rarr 119894119894 ge 0

1198796 le 119894119894 le 1198797

(11)



Minimize Z =119879

sum119894=119905

1205721119903119894 + 1205722119904119894 + 119909119894 (12)

Subject to119879

sum119894=119905

119909119894 le 119868 (13)

119909119894 ge 119890119894minus119903119894 minus 119904119894 + 119889119894 (14)

119909119894 le 119906119903119894minus119903119894 minus 119904119894 + 119889119894 (15)

119903119894 ge 119890119894minus119909119894 minus 119904119894 + 119889119894 (16)

119903119894 le 119906119903119894minus119909119894 minus 119904119894 + 119889119894 (17)

119904119894 ge 119890119894minus119903119894 minus 119909119894 + 119889119894 (18)

119904119894 le 119906119903119894minus119903119894 minus 119909119894 + 119889119894 (19)

119903119905 + 119904119894 + 119909119894 minus 119889119894 ge 119899119894 (20)

119904119879+1 = 0 (21)

1205721 1205722 isin [0 1] (22)

119909119894 119890119894 119903119894 119904119894 119906119903119894 119889119894 119899119894 ge 0

forall119894 isin [119905 119879] (23)

In this model





3 Results




4 Discussion





0

50

100

150

200

250



10

20

30

40

50

60

70

Irrig

atio

n w

ater

(mm

)








0

20

40

60

80

100

120





5 Conclusion












References












































Journal of




Agronomy





Microbiology


























EUR

Evaporation

Transpiration

Irrigation

Precipitation

Evapotranspiration

Drainage

Runoff

URMax



ET119888 = 119896119888 times ET0 (3)














120579119906 = 120579119888119903 minus 120579119901119891 (4)



UR = (120579119888119903 minus 120579119901119891) times 119885

120579119906 = UR119885

(5)






Start

End

YesYes

Yes


T1 le ri le T2

ri + minus di ge ni


si minus di ge ni

Drainage (di)i)


NoNo

No


si

ii = 0












120575119899120575119905

= 120575119903120575119905

+ 120575119904120575119905

+ 120575119894120575119905

minus 120575119889120575119905 (6)


Min119879

sum119894=119905

1205721119903119894 + 1205722119904119894 + 119894119894 (7)




119903119894 isin 119877 997888rarr 119903119894 ge 0

1198791 le 119903119894 le 1198792 forall119894 isin [119905 119879]

119894+2

sum119905=119894


(8)




120579119901119891 lt 119906119903119894 lt 120579119888119903 forall119894 isin [119905 119879] (9)



119904119894 isin 119904 997888rarr 119904119894 ge 0

1198794 le 119904119894 le 1198795 forall119894 isin [119905 119879]

119904119879+1 = 0

(10)







119879

sum119894=119905

119894119894 le 119868

119894119894 isin 119868 997888rarr 119894119894 ge 0

1198796 le 119894119894 le 1198797

(11)



Minimize Z =119879

sum119894=119905

1205721119903119894 + 1205722119904119894 + 119909119894 (12)

Subject to119879

sum119894=119905

119909119894 le 119868 (13)

119909119894 ge 119890119894minus119903119894 minus 119904119894 + 119889119894 (14)

119909119894 le 119906119903119894minus119903119894 minus 119904119894 + 119889119894 (15)

119903119894 ge 119890119894minus119909119894 minus 119904119894 + 119889119894 (16)

119903119894 le 119906119903119894minus119909119894 minus 119904119894 + 119889119894 (17)

119904119894 ge 119890119894minus119903119894 minus 119909119894 + 119889119894 (18)

119904119894 le 119906119903119894minus119903119894 minus 119909119894 + 119889119894 (19)

119903119905 + 119904119894 + 119909119894 minus 119889119894 ge 119899119894 (20)

119904119879+1 = 0 (21)

1205721 1205722 isin [0 1] (22)

119909119894 119890119894 119903119894 119904119894 119906119903119894 119889119894 119899119894 ge 0

forall119894 isin [119905 119879] (23)

In this model





3 Results




4 Discussion





0

50

100

150

200

250



10

20

30

40

50

60

70

Irrig

atio

n w

ater

(mm

)








0

20

40

60

80

100

120





5 Conclusion












References












































Journal of




Agronomy





Microbiology


























Start

End

YesYes

Yes


T1 le ri le T2

ri + minus di ge ni


si minus di ge ni

Drainage (di)i)


NoNo

No


si

ii = 0












120575119899120575119905

= 120575119903120575119905

+ 120575119904120575119905

+ 120575119894120575119905

minus 120575119889120575119905 (6)


Min119879

sum119894=119905

1205721119903119894 + 1205722119904119894 + 119894119894 (7)




119903119894 isin 119877 997888rarr 119903119894 ge 0

1198791 le 119903119894 le 1198792 forall119894 isin [119905 119879]

119894+2

sum119905=119894


(8)




120579119901119891 lt 119906119903119894 lt 120579119888119903 forall119894 isin [119905 119879] (9)



119904119894 isin 119904 997888rarr 119904119894 ge 0

1198794 le 119904119894 le 1198795 forall119894 isin [119905 119879]

119904119879+1 = 0

(10)







119879

sum119894=119905

119894119894 le 119868

119894119894 isin 119868 997888rarr 119894119894 ge 0

1198796 le 119894119894 le 1198797

(11)



Minimize Z =119879

sum119894=119905

1205721119903119894 + 1205722119904119894 + 119909119894 (12)

Subject to119879

sum119894=119905

119909119894 le 119868 (13)

119909119894 ge 119890119894minus119903119894 minus 119904119894 + 119889119894 (14)

119909119894 le 119906119903119894minus119903119894 minus 119904119894 + 119889119894 (15)

119903119894 ge 119890119894minus119909119894 minus 119904119894 + 119889119894 (16)

119903119894 le 119906119903119894minus119909119894 minus 119904119894 + 119889119894 (17)

119904119894 ge 119890119894minus119903119894 minus 119909119894 + 119889119894 (18)

119904119894 le 119906119903119894minus119903119894 minus 119909119894 + 119889119894 (19)

119903119905 + 119904119894 + 119909119894 minus 119889119894 ge 119899119894 (20)

119904119879+1 = 0 (21)

1205721 1205722 isin [0 1] (22)

119909119894 119890119894 119903119894 119904119894 119906119903119894 119889119894 119899119894 ge 0

forall119894 isin [119905 119879] (23)

In this model





3 Results




4 Discussion





0

50

100

150

200

250



10

20

30

40

50

60

70

Irrig

atio

n w

ater

(mm

)








0

20

40

60

80

100

120





5 Conclusion












References












































Journal of




Agronomy





Microbiology































119879

sum119894=119905

119894119894 le 119868

119894119894 isin 119868 997888rarr 119894119894 ge 0

1198796 le 119894119894 le 1198797

(11)



Minimize Z =119879

sum119894=119905

1205721119903119894 + 1205722119904119894 + 119909119894 (12)

Subject to119879

sum119894=119905

119909119894 le 119868 (13)

119909119894 ge 119890119894minus119903119894 minus 119904119894 + 119889119894 (14)

119909119894 le 119906119903119894minus119903119894 minus 119904119894 + 119889119894 (15)

119903119894 ge 119890119894minus119909119894 minus 119904119894 + 119889119894 (16)

119903119894 le 119906119903119894minus119909119894 minus 119904119894 + 119889119894 (17)

119904119894 ge 119890119894minus119903119894 minus 119909119894 + 119889119894 (18)

119904119894 le 119906119903119894minus119903119894 minus 119909119894 + 119889119894 (19)

119903119905 + 119904119894 + 119909119894 minus 119889119894 ge 119899119894 (20)

119904119879+1 = 0 (21)

1205721 1205722 isin [0 1] (22)

119909119894 119890119894 119903119894 119904119894 119906119903119894 119889119894 119899119894 ge 0

forall119894 isin [119905 119879] (23)

In this model





3 Results




4 Discussion





0

50

100

150

200

250



10

20

30

40

50

60

70

Irrig

atio

n w

ater

(mm

)








0

20

40

60

80

100

120





5 Conclusion












References












































Journal of




Agronomy





Microbiology



























0

50

100

150

200

250



10

20

30

40

50

60

70

Irrig

atio

n w

ater

(mm

)








0

20

40

60

80

100

120





5 Conclusion












References












































Journal of




Agronomy





Microbiology





























References












































Journal of




Agronomy





Microbiology







































Journal of




Agronomy





Microbiology

























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