ABSTRACT: This study presents three optimization techniques foron-farm irrigation scheduling in irrigation project planning: namelythe genetic algorithm, simulated annealing and iterative improve-ment methods. The three techniques are applied to planning a394.6 ha irrigation project in the town of Delta, Utah, for optimiz-ing economic profits, simulating water demand, and estimating thecrop area percentages with specific water supply and planted areaconstraints. The comparative optimization results for the 394.6 hairrigated project from the genetic algorithm, simulated annealing,and iterative improvement methods are as follows: (1) the seasonalmaximum net benefits are $113,826, $111,494, and $105,444 perseason, respectively; and (2) the seasonal water demands are3.03*103 m3, 3.0*103 m3, and 2.92*103 m3 per season, respectively.This study also determined the most suitable four parameters ofthe genetic algorithm method for the Delta irrigated project to be:(1) the number of generations equals 800, (2) population size equals50, (3) probability of crossover equals 0.6, and (4) probability ofmutation equals 0.02. Meanwhile, the most suitable three parame-ters of simulated annealing method for the Delta irrigated projectare: (1) initial temperature equals 1,000, (2) number of moves equal90, and (3) cooling rate equals 0.95.(KEY TERMS: genetic algorithm; simulated annealing; iterativeimprovement; optimization; irrigation planning.)
Kuo, Sheng-Feng, Chen-Wuing Liu, and Shih-Kai Chen, 2003. ComparativeStudy of Optimization Techniques for Irrigation Project Planning. J. of theAmerican Water Resources Association (JAWRA) 39(1):59-73.
INTRODUCTION
Irrigation planners must analyze complex climate-soil-plant relationships and apply mathematical opti-mization techniques to determine optimally beneficialcrop patterns and water allocations. A computerbased model which simulates the climate-soil-plant
systems with a novel mathematical optimization tech-nique could help irrigation planners make sound deci-sions before each crop season.
Recently, external influences such as environmen-tal concerns and global trade have been creating newchallenges for the agricultural engineers. Precisionagriculture (PA) is a management strategy that usesinformation technologies to bring data from multiplesources to bear on decisions associated with crop pro-duction (NRC, 1997). Agricultural engineers usinginformation technologies such as genetic algorithm(GA) and simulated annealing (SA) methods will playan increasingly important role in natural resourcemanagement and crop production to meet the newchallenges in the 21st Century (NRC, 1997). There-fore, this preliminary study used the GA and SAmethods to solve the linear programming problem foroptimizing the net benefit of on-farm irrigated projectand the uncertainty expected when extending the on-farm irrigation scheduling to more complicated waterresources management problems in the future, suchas: (1) optimizing combinations of surface and groundwater for irrigation applications, and (2) extendingon-farm irrigation scheduling to optimize reservoiroperation to conserve water resources. Furthermore,the traditional optimization method, iterativeimprovement, was used to compare the GA and SAmethods to distinguish the results from different ran-dom search procedures.
Many models (Hill et al., 1982; Keller, 1987; Smith,1991; Prajamwong, 1994) simulate on-farm irrigationwater demands based on climate-soil-plant systems.The traditional optimizing irrigation planning model
1Paper No. 00134 of the Journal of the American Water Resources Association. Discussions are open until August 1, 2003.2Respectively, Associate Professor, Department of Leisure Management and Graduate Institute of Resource and Environment Manage-
ment, Leader College, Tainan, Taiwan 709, ROC; Professor, Department of Bioenvironmental System Engineering, National Taiwan Universi-ty, No. 1, Section 4, Roosevelt Road, Taipei, Taiwan 106, ROC 10617; and Researcher, ChiSeng Water Management R&D Foundation, 2F, No.39, Lane 9, Section 4, Mu-Cha Road, Taipei, Taiwan, ROC (E-Mail/Liu: [email protected]).
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JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATIONFEBRUARY AMERICAN WATER RESOURCES ASSOCIATION 2003
COMPARATIVE STUDY OF OPTIMIZATION TECHNIQUESFOR IRRIGATION PROJECT PLANNING1
Sheng-Feng Kuo, Chen-Wuing Liu, and Shih-Kai Chen2
seeks to optimize the values to satisfy the objectivefunction and constraints. Traditional optimizationmodels have received extensive attention in irrigationplanning (Lakshminarayana and Rajagopalan, 1977;Maji and Heady, 1978; Matanga and Marino, 1979;Jesus et al., 1987; Paudyal and Gupta, 1990; Ramanet al., 1992). Jesus et al. (1987) developed a linearoptimization model for managing Irrigation DistrictNo. 38 in Sonora, Mexico. Meanwhile, Paudyal andGupta (1990) employed a multilevel optimizationtechnique to resolve the complex problem of irrigationmanagement in a large heterogeneous basin. Further-more, Raman et al. (1992) presented a linear pro-gramming model to generate optimal croppingpatterns based on data tracking previous droughts. Areview of the literature reveals that traditional opti-mization methods have limitations in finding nearglobal optimization results and are difficult to applyto complex irrigation planning problems since theyseek the optimization by searching point to point.Meanwhile, more recently proposed optimizationmethods, such as the genetic algorithm and the simu-lated annealing method, search the entire populationinstead of moving from one point to the next and thusmay overcome the limitations of traditional methods.
A genetic algorithm (GA) is a search procedure thatuses random choice as an effective means of directinga highly exploitative search through a numerical cod-ing of a given parameter space (Goldberg, 1989). Agenetic algorithm has been applied to several opti-mization problems (Wang, 1991; Wentzel et al., 1994;Fahmy et al., 1994; McKinney and Lin, 1994; Reddy,1996; Montesinos et al., 2001). Wang (1991) proposeda genetic algorithm for function optimization andapplied it to the calibration of a conceptual rainfall-runoff model. Wentzel et al. (1994) used GAs to opti-mize a pipe network pumping strategy at New MexicoState University. Fahmy et al. (1994) used GAs foreconomic optimization of river management. Accord-ing to the results of that study, GAs generated nearoptimum solutions for large and complex waterresource problems more efficiently than dynamic pro-gramming techniques. Meanwhile, McKinney and Lin(1994) incorporated GAs into a ground water simula-tion model to solve three ground water managementproblems: (1) maximum pumping from an aquifer, (2)minimum cost in water supply development, and (3)minimum cost in aquifer remediation. Furthermore,Reddy (1996) developed a nonlinear optimizationmodel based on genetic algorithms for land gradingdesign of irrigation fields. Additionally, Montesinos etal. (2001) designed a seasonal furrow irrigation modelwith genetic algorithm to determine a quasi optimumirrigation season calendar based on economic profitmaximization.
On the other hand, simulated annealing (SA) is astochastic computational technique derived from sta-tistical mechanics for finding near global solutions to large optimization problems (Davis,1991). Severalworks have applied simulated annealing to waterresource management (Dougherty and Marryott,1991; Marryott et al., 1993; Mauldon et al., 1993) andirrigation scheduling (Walker, 1992). Dougherty andMarryott (1991) applied simulated annealing to fourproblems of optimal ground water management: (1) adewatering problem, (2) a dewatering problem withzooming, (3) a contamination problem, and (4) con-taminant removal with a slurry wall. Walker (1992)applied the simulated annealing method to a peanutgrowth model to optimize irrigation scheduling. Thepeanut growth model was first applied to determinethe number of irrigation days and the amount of irri-gation during the season. Later, simulated annealingwas implemented in the peanut model.
This study first adopts an irrigation simulation andplanning model to simulate the on-farm surface irri-gation system and obtain the water demand and cropyield information. Three optimization methods, name-ly genetic algorithm, simulated annealing, and inter-active improvements are then applied to maximizethe net benefit of a sample irrigation project in Utah.Various constraints, including minimum and maxi-mum planted crop area and an upper limit on thewater supply, are imposed to satisfy the field condi-tions. Results obtained from the three optimizationmethods are then evaluated to determine the besttechnique for optimizing economic benefits, determin-ing the ideal crop area for a given water supply, andgenerally assisting irrigation planner in makingsound decisions before each crop season.
METHODS
Irrigation Simulation and Planning Model
The irrigation simulation and planning model isdominated by six basic modules: (1) the main module,which directs the running of the model with pull-down menus ability; (2) the data module, for dataentry via a user friendly interface; (3) the weathergeneration module, which generates the daily weath-er data; (4) the on-farm irrigation scheduling module,which simulates the daily water requirements andrelative crop yields; (5) the three optimization tech-niques module, which optimize the project maximumbenefit; and (6) the results module, which presentsresults using tables, graphs, and printouts and subse-quently sends these results to the three optimization
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models for economic optimization and determiningthe optimal patterns and irrigation water application.
Figure 1 presents the framework and logicemployed in the farm irrigation simulation and plan-ning model. As presented, the model requires sixbasic types of data: (1) project site and operation data,(2) command area data, (3) seasonal water supplydata, (4) monthly weather data, (5) soil propertiesdata, and (6) crop phenology and economic data. Theweather generation module from CADSM (Prajam-wong, 1994) is adopted herein to generate daily refer-ence crop evapotranspiration and rainfall data byusing the monthly mean and standard deviationsdata.
On-Farm Irrigation Scheduling
The daily on-farm irrigation scheduling modulesimulates the on-farm water balance and estimatesthe relative crop yield and irrigation water require-ments using the basic project data and generateddaily weather data. Within the module, the relativecrop yield is influenced by two factors: (1) the water
stress due to insufficient water for crop evapotranspi-ration, and (2) waterlogging due to infiltration,produced by overirrigation and/or precipitation.Although the percentage of relative crop yield startsat 100 percent at the beginning of a growing season,the value can be reduced to less than 100 percent ifthere is any water stress or waterlogging during thegrowing season.
The relative yield reduction due to water stress iscalculated at the end of each growth stage based onthe ratio of cumulative potential crop evapotranspira-tion ETc,stage and actual crop evapotranspirationETca,stage in each stage. The relationships can bedescribed by Equation (1) (Prajamwong, 1994)
where Yam,stage denotes the relative yield reductiondue to water stress at each stage; Ky,stage representsthe crop yield response factor at the same stage;ETca,stage is the actual crop evapotranspiration at theend of the stage; and ETc,stage denotes the potentialcrop evapotranspiration at the end of the stage.
The minimum value of Yam,stage at each growthstage was chosen to be representative of the relativeyield reduction due to water stress over the entireseason Yam,season as given by Equation (2)
Yam,season = Min (Yam,1; Yam,2; . . . ; Yam,stage)
The relative yield reduction due to waterlogging iscalculated at the end of the season based on the ratioof cumulative total infiltration fseason and the maxi-mum net depletable depth dn in the root zone. Theserelationships can be represented by Equations (3) and(4) (Prajamwong, 1994)
dn = DmaxMARz
where Ya,season denotes the relative yield reductiondue to infiltration over the entire season; a is theempirical coefficient; Dmax represents the maximumallowable depletion (fraction); MA is the available soilmoisture in mm/m; and Rz denotes the maximum rootdepth in m.
The product of relative yield reduction due to waterstress over the entire season Yam,season and relativeyield reduction due to waterlogging over the entireseason Ya,season is the final value of relative crop yieldat the end of the growing season.
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Figure 1. The Framework of the Irrigation and Planning Model.
Y KET
ETam stage y stageca stage
c stage, ,
,
,= − −
1 1 (1)
(2)
Y af
da seasonseason
n, = −
1 (3)
(4)
The output from this module includes relative cropyield and crop irrigation water requirements. Bothoutputs are required as inputs for optimization usingthe following three optimization modules: (1) thegenetic algorithm, (2) simulated annealing, and (3)iterative improvement methods.
Genetic Algorithm
The genetic algorithm module has been applied tothe on-farm irrigation scheduling module to maximizethe project benefit. The computational procedures ofthe genetic algorithm modules divide into the follow-ing steps: (1) receive input data, (2) user interface toenter genetic algorithms, (3) define chromosome torepresent problem, (4) decode chromosome into realnumber, (5) constraints control, (6) fitness value fromobject function, and (7) three simple genetic algorithmoperators – reproduction, crossover, and mutation.
To design the length of a chromosome to representan irrigation project, the total number of crops withineach command area is first calculated. Each crop isthen assigned seven binary digits to represent itsarea, which can range from 1 to 100 percent of thetotal area of each command area (seven binary digitsgive a value of 0 to 27-1, or 0 to 127 when expresseddecimally). Finally, the length of a chromosome equalsthe total number of crop types multiplied by seven.The model is tested using two command areas in theDelta, Utah, irrigation project. The first commandarea, UCA No. 2, includes three crop types, while thesecond command area, UCA No. 4, includes four croptypes. Therefore, these two command areas containseven crop types, and thus the length of a chromo-some should be 7 * 7 = 49. For a chromosome string of49 binary digits, the seven crops in the two Delta,Utah, command areas can be represented in codedform, as in Figure 2.
The chromosome can be decoded into a decimalnumber to represent the crop area within each com-mand area. The conventional decoding method is usedherein. Consider a problem with k decision variables
xi, i = 1, 2, ..., I, defined with the intervals xi ∈[ai, bi].Each decision variable can be decoded as a binarysubstring of length mi, and the decoded decimal, xi,can be obtained from Equation (5) (McKinney andLin, 1994)
where s is the summation identifier for substringlength and mi is the substring length.
The irrigation project, with planted area of 394.6hectares, in the following case study contains sevencrop types. Therefore, this problem has seven decisionvariables (xi), and I equals seven. Without consideringinherent crop area constraints, the percentage area ofeach crop type can range from 1 to 100 percent of thetotal command area. Therefore, the interval for eachdecision variable can be represented as xi ∈[1, 100],where ai equals 1 and bi equals 100. In conclusion,Equation (1) can decode the binary digits into anactual number ranging between 1 and 100. The nextstep is to transform this decimal number into a croparea percentage, Areaj,%, and area, Areaj,ha, withineach command area. A simple averaging techniquewas used, as presented by Equations (6) and (7).
where j is the crop index; Nc denotes the number ofcrops within each command area; and Auca representsthe area of each unit command area.
Simulated Annealing
SA is another optimization method implementedwith the on-farm irrigation scheduling module tomaximize project benefits. SA computation involvesthe following steps: (1) receive output from the irriga-tion module, (2) enter the parameters required bysimulated annealing, (3) define the chromosomesaccording to the problem, (4) generate a random ini-tial chromosome, (5) decode the chromosome into areal number, (6) apply constraints, (7) apply the objec-tive function and fitness value, (8) implement theannealing schedule using Boltzmann probability, and(9) set the cooling rate and termination criterion.
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Figure 2. A Sample Chromosome Coding Scheme forRepresenting Seven Crops in the Delta Project.
x ab a
bi ii im s
s
ms
i
i
= + −−
×=∑
2 12
0
(5)
Ax
xj
j
jj
Nc,% =
=∑
1
100
AA
Aj haj
uca,,%=
100
(6)
(7)
Annealing scheduling is the key to the simulatedannealing method. This procedure is one of the major differences between simulated annealing andtraditional optimization methods (e.g., the IterativeImprovement or Monte Carlo methods) and allowsperturbations to move uphill in a controlled fashion.Annealing scheduling involves the following steps:
1. Simulated annealing allows numerous moveswithin one temperature value, and thus the first stepis to compare the energy difference or project benefitin dollars, ∆E, from previous moves to the currentmove, ∆E = Emove - Emove+1.
2. If the energy difference is negative (∆E < 0), themaximum benefit of the irrigation project and relatedcrop areas are accepted at this move because theenergy or economical benefit increases between theprevious and current moves.
3. If the energy difference is positive (∆E > 0), noenergy improvement occurred, but the irrigation pro-ject maximum benefit and related crop areas can stillupdate if the Boltzmann probability, Pr, exceeds thegenerated uniform random number, r. The Boltzmannprobability can be defined as in Equation (8)
The above equation reveals that Pr increases withtemperature, Tsa; furthermore, the system has moreopportunity to update the parameter if ∆E greaterthan 0. This feature also implies that higher tempera-tures increase the ability of the system to rearrangethe atoms (that is to jump away from local optima)and optimize the results. As the temperature contin-ues to decrease, the system stabilizes because the Prvalue is small, and the parameter can no longer beupdated if ∆E greater than 0. Finally, this procedurecan determine the global (or near global) optimum.
Iterative Improvement Method
For comparison with the previous two global opti-mization methods (namely the genetic algorithm andsimulated annealing methods), one of the traditionaloptimization methods, the iterative improvementmethod, was employed herein.
Kirkpatrick (1984) mentioned that the iterativeimprovement method is a special case of simulatedannealing. As discussed above, the annealing schedul-ing process in the simulated annealing method isa major difference between this method and thetraditional optimization method. Without parallel
searching, as in the genetic algorithm method, orannealing scheduling, as in the simulated annealingmethod, the iterative improvement method is fre-quently trapped at a local minimum (or maximum).This is because this method only accepts those moveswhich improve the energy or benefit. Therefore, theiterative improvement method should not provideresults as good as those from the genetic algorithmand simulated annealing methods.
Objective Function and Fitness Value
Herein, the objective function includes the incomefrom crop harvest, as well as irrigation and crop pro-duction costs. The objective is to maximize the benefitof the irrigation project to the seven crops growing inthe two command areas. Within the loop calculatingchromosome size, the objective function returns a fit-ness value to the model and then updates the fitnessvalue and related crop allocated area whenever thisvalue increased. The fitness value at the end of thechromosome loop is the most beneficial value withinthe loop. Furthermore, the maximum fitness value isselected from the generation number loop. Thus, thecalculations optimize the fitness value and relatedcrop area. The objective function can be mathemati-cally expressed as in Equation (9)
Maximize:
where F is the net profit in $/season for the irrigatedproject; i,j is the command area and crop index,respectively; N is the number of command area withinirrigated project; Nc is the number of crops withineach command area; Pi,j is unit price of the jth crop inthe ith command area in $/ton; Yi,j is yield per hectareof jth crop in the ith command area in ton/ha; Si,j isseed cost per hectare of jth crop in the ith commandarea in $/ha; Fi,j is fertilizer cost of jth crop in the ithcommand area in $/ha; Li,j is labor cost of jth crop inthe ith command area in $/ha; Oi,j is operation cost ofjth crop in the ith command area in $/ha; Ai,j is plant-ed area of jth crop in the ith command area in ha; W isunit price of irrigation water in $/m3; and Qi,j iscumulative water requirement of jth crop in the ithcommand area in m3.
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P erE Tsa= −∆ / (8)
F P Y S F L O
A W Q
i j i j i j i j i j i jj
N
i
N
i j i jj
N
i
N
c
c
= − − − −( )
−
==
==
∑∑
∑∑
max , , , , , ,
, ,
11
11
(9)
The objective function is subject to the followingconstraints
1. To consider social factors and prevent a singlehigh-value crop from dominating the search for maxi-mum benefit, maximum and minimum area percent-ages for the crops must be considered
where Aminj,% and Amax
j,% are the minimum and maximumpercentage area values of crop j in command area i inpercent, respectively.
2. The cumulative water demand of crop j in com-mand area i should not exceed the water supply avail-able for each command area
where Qdem denotes the irrigation water requirementfor crop j in command area i in m3; and Qsup repre-sents the available water supply for command area iin m3.
CASE STUDY
Site Description
The Wilson Canal System, which is close to the cityof Delta in central Utah, was used herein. The WilsonCanal System, which is one of numerous diversions inthe Sevier River Basins, is operated by the AbrahamIrrigation Company. The Wilson Canal System is anon-demand irrigation system with a good communica-tions network. The Wilson Canal is 11,480 m long,and its water is sourced from the Gunnison BendReservoir. Figures 3 and 4 present the location of theSevier River Basin and the plan view of the commandarea within the irrigated project in Utah, respectively.
The climate in the Delta area is essentially a colddesert type, arid with cold winters and warm sum-mers. The UCA No. 2 and UCA No. 4 command areaswithin the Wilson Canal System were irrigated by on-demand irrigation method and selected for evaluatingthe model. The UCA No. 2 command area has a 2,896m water course length, a planted area of 83.3ha, andthree crop types – alfalfa, barley, and corn. Mean-while, the UCA No. 4 command area has a 12,350 mwater course, a planted area of 311.3ha, and four croptypes – alfalfa, barley, corn, and wheat.
Based on weather data from 1993 for Delta, Utah,Figure 5 summarizes the daily simulation resultsfrom the on-farm irrigation scheduling module todemonstrate the relationship between soil moisture,irrigation depth, and rainfall for the alfalfa and bar-ley crops in the UCA No. 4 command area. Further-more, Tables 1 and 2 list the seasonal outputs for theUCA No. 2 and UCA No. 4 command areas, respec-tively.
Application of the Genetic Algorithm
To test the effectiveness of the GA work, the fourparameters are set as follows: (1) the number of
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A A Aj j j,%min
,% ,%max≤ ≤ (10)
Q Qdemj
NC
=∑ ≤
1sup (11)
Figure 3. Sevier River Basin, Utah (Tzou, 1989).
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Figure 4. Plan View and Crop Types of the UCA No. 4 Command Area Within Delta, Utah, Irrigation Project.
generations equals 100, (2) the population size is 50,(3) the probability of crossover (Pc) equals 0.6, and (4)and probability of mutation (Pm) is 0.02. Figure 6 pre-sents a sample graph from the GA method to repre-sent the benefit of the Delta project during searching.Figure 7 illustrates the relationship between theaverage, standard deviation, and maximum benefitvalues for different numbers of generations after run-ning the model. SGA performs relatively well, withthe average of fitness values tending to increase from$88,292 to $139,113 as the number of generationincreases. On the other hand, the standard deviationof the fitness decreases from $17,127 to $3,291. Themaximum benefit is updated from generation to nextgeneration if the benefit improves. Notably, the maxi-mum and average fitness values have nearly thesame increasing tendency with generation number.
Extensive testing reveals that the most appropriateparameters for this work are as follows: (1) number
of generations equal to 800, (2) population size of 50,(3) probability of crossover equals to 0.6, and (4) prob-ability of mutation of 0.02. Table 3 summarizes thefinal results for these parameters following ten runs.Each run appears to have succeeded because the ben-efit, water demand, and related crop allocated areasare quite close. As presented, the maximum benefit isup to $114,734 at Run 3, and standard deviation ofthe benefit from the ten runs ranges from a very lowfigure to a maximum of $646. Therefore, using theabove parameters creates a good chance of obtainingnear global optimal values for this irrigated projectplanning problem. The average values shown in Table3 can be considered the optimal plan for Delta, Utah.
Application of the Simulated Annealing
The simulated annealing method requires threeparameters: (1) initial temperature, (2) number ofmoves, and (3) cooling rate. Based on many tests, themost appropriate parameters for the SA method here-in are as follows: (1) an initial temperature equal to1,000, (2) a number of moves equal to 90, and (3) acooling rate equal to 0.95. Table 4 summarizes thefinal results for these parameters from ten runs.Each run appears to have succeeded because there islittle variation in the benefit, water demand, andrelated crop allocated areas. As presented, the maxi-mum benefit was $114,857 at Run 2. The averagebenefit was $114,494 with a standard deviation of$1,581. The average values as listed in Table 4 can beconsidered to represent the optimal irrigation plan forDelta, Utah. Figure 8 displays the sample graph fromthe SA method to represent the benefit of the Deltaproject during the searching process. Notably, thegraphs are read from right to left because the anneal-ing proceeds from high to low temperatures. At highertemperatures, the annealing scheduling makes thegraph move either down or up, while at lower temper-atures the graphs tend toward equilibrium and reachnear global optimum results.
Application of the Iterative Improvement
Iterative improvement is the third method and canbe recognized as the traditional optimization methodthat is applied. This method does not require anyparameters, and the annealing schedule is the maindifference between this method and the simulatedannealing method. Thus, the iterative improvementmethod cannot update the parameters like the simu-lated annealing method can, as the energy or benefitmay not improve from the previous iteration.
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Figure 5. The Relationship Between Soil Moisture, IrrigationDepth, and Rainfall for Corn and Wheat Crops
in the UCA No. 4 Command Area.
Because of significant differences in benefit amongruns, 50 runs were conducted to test the iterativeimprovement method and more accurately assess theresults produced by its simulation results. Table 5summarizes the final results from the 50 runs. Aspresented, the average benefit was $105,444 with astandard deviation of $2,911. More interestingly, themaximum benefit, $113,127, appears at the 15th run,and this value approaches the near global optimalresults. On the other hand, the minimum benefitoccurs in the first run and is as low as $99,963.
Figure 9 illustrates sample graphs from the 50runs to represent the relationship between the projectbenefit and iterations during searching. The curvesare shaped like the previous graphs from the geneticalgorithm method because the net benefit is updatedonly if its values improved from one iteration to thenext. The graphs also reveal that the benefit initiallyincreases quickly (that is, within the 50 iterations),then climbs slowly to reach its final values as thenumber of iterations increases. However, in contrastto the genetic algorithm method, different iterations
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TABLE 2. Seasonal Outputs for the UCA No. 4 Command Area From the On-Farm Irrigation Scheduling Submodel.
Alfalfa Barley Corn Wheat
Potential ET (mm/season) 1039.3 572.1 523.4 611.2Actual ET (mm/season) 906.2 528.7 469.6 558.1Evaporation From Wet Soil Surface (mm/season) 3.4 37.9 21.9 34.4Number of Irrigations 7 6 4 6Total Irrigation Depth (mm/season) 1039.5 531.4 490.9 539.4Deep Percolation (mm/season) 68.3 35.1 38.4 35.6Surface Runoff (mm/season) 27.8 14.3 15.6 14.5
TABLE 1. Seasonal Outputs for the UCA No. 2 Command Area From the On-Farm Irrigation Scheduling Submodel.
Alfalfa Barley Corn
Potential ET (mm/season) 1038.0 555.6 514.8Actual ET (mm/season) 907 505.7 460.9Evaporation From Wet Soil Surface (mm/season) 2.1 21.4 13.4Number of Irrigations 6 4 3Total Irrigation Depth (mm/season) 1067.9 441.7 471.8Deep Percolation (mm/season) 70.1 29.4 37.2Surface Runoff (mm/season) 28.5 11.9 15.1
Figure 7. The Relationship Between Maximum, Average, andStandard Deviation of Fitness, and Number of Generations for GA.
Figure 6. Sample Graphs for the GA Method WithPopulation Sizes of 50, Pc of 0.6, and Pm of 0.02.
are performed on each run because the stop criterionis defined as 50 consecutive iterations during whichnet benefit remains unchanged. This criterion wasbased on numerous test results. The iterative
improvement method merely searches from one pointto another and lacks the parallel searching functionas genetic algorithms and simulated annealing meth-ods.
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TABLE 3. Genetic Algorithm Results With Population Size of 50, CrossoverProbability of 0.6, and Mutation Probability of 0.02.
Project UCA No. 2 UCA No. 4Net Water Water Water
Benefit Demand Demand Demand($1,000/ (1,000 m3/ Alfalfa Barley Corn (1,000 m3/ Alfalfa Barley Corn Wheat (1,000 m3/
Run season) season) (%) (%) (%) season) (%) (%) (%) (%) season)
01 114.416 3046.018 71.85 08.89 19.26 747.581 41.89 39.19 15.32 3.60 2298.437
02 113.144 3007.601 66.67 18.94 14.39 719.313 40.83 44.95 10.09 4.13 2288.288
03 114.734 3037.446 71.46 19.05 09.52 742.931 41.40 41.94 12.37 4.30 2294.515
04 113.447 3015.969 68.09 22.34 09.57 725.503 40.58 46.86 5.80 6.76 2290.465
05 114.635 3039.702 72.14 22.14 05.71 745.702 42.31 30.77 23.93 2.99 2294.000
06 114.170 3032.308 70.31 17.97 11.72 737.660 41.40 41.40 12.37 4.84 2294.648
07 114.044 3039.927 71.43 15.08 13.49 743.926 41.77 37.13 16.03 5.06 2296.000
08 112.826 3024.938 71.01 21.74 07.25 740.200 41.85 26.87 25.99 5.29 2284.738
09 113.773 3018.331 67.12 23.29 09.59 720.490 41.56 41.56 11.93 4.94 2297.842
10 113.070 3033.697 71.00 21.00 08.00 740.313 41.08 38.59 10.37 9.96 2293.384
Average 113.826 3029.594 70.11 19.04 10.85 736.362 41.47 38.93 14.42 5.49 2293.232
Maximum 114.734
Minimum 112.826
Stand. Dev. 000.646
TABLE 4. Simulated Annealing Results With Initial Temperature of 1,000,Number of Moves of 90, and Cooling Rate of 0.95.
Project UCA No. 2 UCA No. 4Net Water Water Water
Benefit Demand Demand Demand($1,000/ (1,000 m3/ Alfalfa Barley Corn (1,000 m3/ Alfalfa Barley Corn Wheat (1,000 m3/
Run season) season) (%) (%) (%) season) (%) (%) (%) (%) season)
01 112.715 3017.371 67.19 17.97 14.84 722.143 42.65 27.21 27.21 2.94 2295.228
02 114.857 3048.221 72.90 21.50 5.61 749.610 42.67 28.89 24.89 3.56 2298.612
03 109.517 2990.306 70.69 5.17 24.14 742.742 39.81 26.85 29.63 3.70 2247.564
04 110.593 2999.315 64.66 27.82 07.52 707.130 42.54 19.74 29.39 8.33 2292.185
05 112.255 3014.310 72.55 13.73 13.73 749.830 39.27 43.98 9.95 6.81 2264.481
06 110.005 3024.033 71.00 12.00 17.00 742.571 42.62 10.13 38.82 8.44 2281.462
07 112.646 3017.384 68.87 19.81 11.32 730.025 41.05 37.89 14.21 6.84 2287.359
08 110.144 2957.283 58.39 35.77 05.84 674.015 41.92 26.26 27.78 4.04 2283.267
09 110.318 2980.368 68.57 07.62 23.81 731.610 38.15 48.55 08.09 5.20 2248.758
10 111.886 2974.774 60.51 29.94 09.55 685.982 41.48 38.07 17.61 2.84 2288.793
Average 114.494 3002.337 67.53 19.13 13.34 723.566 41.22 30.76 22.76 5.27 2278.771
Maximum 114.857
Minimum 109.517
Stand. Dev. 001.581
Comparison of Three Optimization Methods
Two global optimization methods – genetic algo-rithm and simulated annealing – and one traditionaloptimization method were used herein to determinethe maximum benefit, water demand, and relatedcrop area percentages for Delta, Utah. This sectioncompares and discusses the three optimization meth-ods used in the irrigated project planning, seeking themethod that maximized benefit while minimizingwater demand.
From previous sections on each optimizationmethod, Tables 3, 4, and 5 represent the final resultsfrom the genetic algorithm, simulated annealing, anditerative improvement methods, respectively. Mean-while, Table 6 summarizes the results from the threeoptimization methods. The contents of Table 6 are dis-cussed below.
1. The crop area percentages and water demandfall within the constraints for all three optimizationmethods. Therefore, all three methods effectively han-dle the constraints that are subject to the objectivefunction.
2. Column 2 illustrates that the maximum benefitsof the genetic algorithm and simulated annealingmethods are $113,826 and $111,494, respectively. Themaximum benefits from the genetic algorithm andsimulated annealing methods are very close, and thusboth methods find the near optimum benefit. On theother hand, the maximum benefit from the iterativeimprovement method is only $105,444. Clearly, thetraditional optimization method (that is, iterativeimprovement) can only sometimes find the local
optimum values, while the global optimization meth-ods consistently find near optimal values.
3. Consider the crop area percentages in columns 5to 8 for the three optimization methods. All threemethods have the same trend to search the areas ofcrops within the UCA No. 2 and UCA No. 4 commandareas. That is, the order of crop portions from large tosmall is Alfalfa > Barley > Corn in the UCA No. 2command area and Alfalfa > Barley > Corn > Wheatin the UCA No. 4 command area. Table 6 also showsthat the crop area percentages from the genetic algo-rithm and simulated annealing methods for the UCANo. 2 and UCA No. 4 command areas produce verysimilar results in terms of maximum benefit. On theother hand, the crop area percentages using the itera-tive improvement method for UCA No. 2 differ fromthe GA and SA methods.
4. Consider the standard deviation in column 1 forthe three methods. The standard deviations areselected from the most suitable data set for the GAand SA methods and are calculated based on the max-imum benefit following many runs (i.e., 10 runs forthe GA and SA methods, and 50 runs for the iterativeimprovement method). The GA method revealed tohave the lowest standard deviation, of 0.646, followedby 1.581 for the SA method. Meanwhile, the iterativeimprovement method has the highest standard devia-tion, of 2.911.
5. Regarding the water demand for Delta, Utah,displayed in column 4, water requirements rank asfollows: iterative improvement method (2,922,927m3) < SA method (3,002,337 m3) < GA method(3,029,594 m3).
JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION 69 JAWRA
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Figure 8. Sample Graphs for the Simulated Annealing Method. Figure 9. Sample Graphs for the Iterative Improvement Method.
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TABLE 5. Summary Results From the Iterative Improvement Method.
Project UCA No. 2 UCA No. 4Net Water Water Water
Benefit Demand Demand Demand($1,000/ (1,000 m3/ Alfalfa Barley Corn (1,000 m3/ Alfalfa Barley Corn Wheat (1,000 m3/
Run season) season) (%) (%) (%) season) (%) (%) (%) (%) season)
01 105.934 2905.858 49.69 36.81 13.50 630.549 41.18 24.79 25.63 08.40 2275.3092 102.385 2872.980 38.10 21.21 40.69 576.870 40.89 38.22 06.67 14.22 2296.1103 104.126 2991.916 71.54 07.69 20.77 746.325 37.91 23.70 11.85 26.54 2245.5914 105.469 2864.156 43.22 36.18 20.60 598.544 40.08 38.02 18.60 03.31 2265.6125 103.386 2953.990 53.79 18.94 27.27 655.363 42.77 07.23 29.52 20.48 2298.6276 110.138 2962.261 59.70 29.10 11.19 682.178 41.71 27.49 27.49 03.32 2280.0837 102.849 2898.738 62.79 25.58 11.63 698.401 38.10 07.14 46.43 08.33 2200.3378 104.107 2979.569 71.43 21.43 07.14 742.334 38.97 07.04 30.99 23.00 2237.2359 104.541 2913.359 46.02 05.31 48.67 620.198 41.95 29.76 20.98 07.32 2293.162
10 103.084 2917.083 56.25 19.79 23.96 667.375 38.37 30.61 12.65 18.37 2249.70811 104.958 2959.160 64.04 22.81 13.16 705.276 39.52 18.15 23.79 18.55 2253.88412 102.223 2882.396 52.21 33.82 13.97 643.775 39.92 09.24 39.50 11.34 2238.62213 103.202 2943.305 56.41 28.85 14.74 665.900 40.72 14.48 21.27 23.53 2277.40514 113.127 3009.453 66.22 10.81 22.97 719.115 40.85 47.89 08.45 02.82 2290.33815 107.831 2918.232 53.33 36.19 10.48 648.779 39.612 41.55 10.63 08.21 2269.45316 100.694 2919.500 59.38 01.09 19.53 682.566 38.70 10.43 27.83 23.04 2236.93417 104.082 2860.775 36.05 19.77 44.19 567.060 41.56 39.61 14.94 03.90 2293.71518 104.627 2896.474 48.28 17.24 34.48 628.418 40.98 25.82 28.28 04.92 2268.05519 105.336 2934.078 63.64 25.97 10.39 702.502 37.29 33.90 12.99 15.82 2231.57620 107.782 3029.893 71.11 10.37 18.52 743.531 41.90 10.95 28.10 19.05 2286.36221 101.450 2835.903 44.97 15.38 39.64 612.471 37.50 37.89 19.92 04.69 2223.43222 107.132 2928.036 57.99 22.49 19.53 675.330 40.54 22.30 34.46 02.70 2252.70623 106.922 2929.845 51.19 29.17 19.64 639.900 43.00 15.46 36.23 05.31 2289.94424 106.589 2949.033 64.42 07.69 27.88 710.993 39.76 19.68 36.55 04.02 2238.04025 104.340 2893.969 57.41 35.80 06.79 669.107 37.96 25.71 25.71 10.61 2224.86326 103.257 2877.827 59.60 15.15 25.25 685.153 34.81 44.94 11.39 08.86 2192.67427 99.963 2798.687 33.19 37.61 29.20 548.380 39.24 31.65 21.10 08.02 2250.30728 106.892 2937.059 56.00 30.86 13.14 663.358 41.76 16.48 34.07 07.69 2273.70029 107.144 2993.394 62.50 20.83 16.64 698.148 40.93 28.27 09.28 21.52 2295.24530 107.722 2950.164 62.50 16.67 20.83 699.193 38.32 42.52 09.35 09.81 2250.97131 103.593 2934.983 52.69 19.35 27.96 649.798 40.98 22.40 17.49 19.13 2285.18532 112.368 3024.365 72.45 12.24 15.31 749.704 41.45 26.50 28.63 03.42 2274.66133 104.968 2879.432 46.15 41.67 12.18 611.756 40.54 27.70 23.65 08.11 2267.67534 104.995 2912.221 49.43 43.68 06.90 627.496 42.28 12.08 32.89 12.75 2284.72635 108.673 2947.470 66.67 5.80 27.54 722.609 36.88 49.65 10.64 02.84 2224.86136 107.358 2923.261 59.12 32.70 08.18 678.385 38.33 38.75 14.17 08.75 2244.87637 103.578 2946.779 64.62 09.23 26.15 711.562 37.34 31.12 11.62 19.92 2235.21738 106.725 2951.954 65.17 14.61 20.22 712.961 39.11 24.44 28.44 08.00 2238.99339 106.933 2933.514 59.55 21.35 19.10 683.374 38.46 40.00 11.79 09.74 2250.14140 112.238 3005.632 66.90 17.93 15.17 720.707 41.71 31.66 23.62 03.02 2284.92541 102.484 2909.340 47.80 33.96 18.24 621.856 41.46 16.10 21.95 20.49 2287.48442 104.441 2881.469 62.14 24.29 13.57 695.509 35.38 36.32 23.11 05.19 2185.95943 107.283 2933.724 69.23 20.00 10.77 731.779 34.92 50.00 05.56 09.52 2201.94544 103.526 2827.391 38.68 41.98 19.34 574.562 39.76 33.33 24.50 02.41 2252.83045 110.708 2963.566 57.27 35.45 07.27 668.525 42.14 31.45 21.38 05.03 2295.04146 104.999 2858.641 36.69 27.22 36.09 568.368 41.15 43.36 12.39 03.10 2290.27347 101.943 2825.992 48.39 44.09 07.53 622.238 36.00 37.20 17.60 09.20 2203.75348 101.963 2860.571 50.79 09.52 39.68 642.856 37.50 32.87 24.54 05.09 2217.71549 107.172 2959.243 61.04 16.88 22.08 691.884 40.88 22.01 28.30 08.81 2267.35950 104.951 2959.727 55.06 26.58 18.35 659.779 41.59 21.96 14.02 22.43 2299.948
Average 105.444 2922.927 56.06 23.58 20.36 665.456 39.69 28.00 21.62 10.69 2257.471Maximum 113.127Minimum 99.963Stand. Dev. 2.911
Comparison of Search Procedures
All of the three optimization methods have similaryet not identical procedures for maximizing the bene-fit for irrigated project planning.
1. The three methods are similar and are based onthe “random” searching technique. However, the ran-dom searching process differs among the three meth-ods. The iterative improvement method merelyconducts a “blind random” search from point to point.Meanwhile, the genetic algorithm method beginssearching from a random initial condition but is not“blind” because the simple genetic algorithm (SGA)contains three operators – reproduction, crossover,and mutation. The simulated annealing searchingprocedures originates from the iterative improvementmethod, yet annealing scheduling allows the SAmethod to update the parameters even if the energy isno better than in the previous iteration.
2. The GA method requires four input parameters– number of generations, population size, probabilityof crossover, and probability of mutation. The SAmethod requires three parameters – initial tempera-ture, number of moves, and cooling rate. The draw-back of the GA and SA methods is that extensiveexperience and work are needed to determine themost suitable parameters for their applied problems.Since the iterative improvement method has no inputparameters, this method does not suffer the abovedisadvantages.
3. The two global optimization methods – geneticalgorithm and simulated annealing – were compared.
Theoretically, the SA method should converge betterthan the GA method because SA will stop when thefinal solution remains unchanged for many iterations,and thus the final solutions will normally representthe best result. Unlike the SA method, the stop crite-rion for the GA method depends on the number ofgenerations. For example, 800 generations were usedfor all of the tests herein; that is, the GA methodstopped after 800 generations, even though more opti-mal values may have been found if the model hadsearched for more than 800 generations.
CONCLUSION
The genetic algorithm, simulated annealing, anditerative improvement methods were introduced andcompared based on how well they optimized the bene-fits of water and crop management in an irrigationproject. The following conclusions are based on theresults from this irrigation simulation and optimiza-tion model.
1. Genetic algorithms can be effectively and effi-ciently applied to irrigation project planning in opti-mizing economic benefit and determining ideal cropareas for a given water supply.
2. The simulated annealing method performsalmost as well as the genetic algorithm method.Annealing scheduling is a powerful means of allowingthe SA method to find better more optimal resultsthan the iterative improvement method.
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TABLE 6. Summary Results Comparing the Three Optimization Methods.
(1) (2) (3) (4) (5) (6) (7) (8) (9)Standard Net WaterDeviation Benefit Demand
($1,000/ ($1,000/ (1,000 m3/ Alfalfa Barley Corn WheatMethods Items season) season) season) (%) (%) (%) (%)
Genetic Algorithm Project 0.646 113.826 3029.594 – – – –
UCA No. 2 – – 0736.362 70.1 19.0 10.9 –
UCA No. 4 – – 2293.232 41.5 38.9 14.4 5.2
Simulated Annealing Project 1.581 111.494 3002.337 – – – –
UCA No. 2 – – 0723.566 67.5 19.1 13.3 –
UCA No. 4 – – 2278.771 41.2 30.8 22.8 5.3
Iterative Improvement Project 2.911 105.444 2922.927 – – – –
UCA No. 2 – – 0665.456 56.1 23.6 20.4 –
UCA No. 4 – – 2257.471 39.7 28.0 21.6 10.7
3. The standard for choosing the most suitableparameters for GA and SA methods is based on thedata set selected that has the highest average andlowest deviation of benefit from many runs. The mostsuitable four parameters for GA are the number ofgenerations equals 800, population size equal 50,probability of crossover equals 0.6, and probability ofmutation equals 0.02. Meanwhile, the most suitablethree parameters for SA are initial temperatureequals 1000, number of moves equal 90, and coolingrate equals 0.95.
4. The three optimization methods are comparedfrom the perspective of the applicability and efficiencyof the searching procedures. The comparative resultsindicate that both global optimization methods per-form very well, and the results from the project bene-fit, water demand, and crop allocated area are veryclose. The genetic algorithm and the simulatedannealing method both found similar near optimalbenefits for Delta, Utah. On the other hand, the itera-tive improvement method produced worse resultsthan did the GA and SA methods because the benefitswere lower and the standard deviation higher thanfor the GA and SA methods.
5. The genetic algorithm, simulation annealing,and iterative improvement methods have been com-pared and successfully implemented to an on-farmirrigation scheduling submodel for optimizing irriga-tion project benefits. Yet, the objective function andconstraints are linear, and only on-farm irrigationproblems are considered in this study. The geneticalgorithm and simulated annealing methods can cer-tainly be applied to more complicated water resourcesmanagement problems in the future, such as optimiz-ing combinations of surface and ground water forirrigation applications, and extending on-farm irriga-tion scheduling to optimize reservoir operation to con-serve water resources.
LITERATURE CITED
Davis, L., 1991. Handbook of Genetic Algorithms.Van NostrandReinhold, New York, New York.
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NOTATIONS
a empirical coefficient
ai,bi minimum and maximum value of decoded decimal
A crop planted area, ha
Aj, % crop planted area within command area, %
Aj,ha crop planed area within command area, ha
Auca area of each unit command area, ha
Aminj,% minimum percentage area of crop within command
area, %
Amaxj,% maximum percentage area of crop within command
area, %
Dmax soil maximum allowable depletion, mm
dn maximum net depletable depth, mm
Emove project benefit at current move during annealing scheduling, $
ETc,stage potential crop evapotranspiration at each stage,mm/day
ETca,stage actual crop evapotranspiration at each stage, mm/day
∆E change of project benefit from current and previous move, $
fseason cumulative seasonal infiltration, mm
F objective function, $/season
Fi,j fertilizer cost of jth crop in the ith command area, $/ha
i, j command area and crop index
Ky,stage crop yield response factor at current growth stage
mi substring length (dimensionless)
MA available soil moisture, mm/m
N number of command area within irrigated project
Nc number of crop within command area
Oi,j operation cost of jth crop in the ith command area, $/ha
Pi,j unit price of the jth crop in the ith command area, $/ha
Pr Boltzmann probability
Qdem cumulative crop water demand in command area, m3
Qsup available water supply for command area, m3
Qi,j cumulative water requirement of jth crop in the ith
command area, m3
Rz root depth, mm
s summation identifier for substring length
Si,j seed cost per hectare of jth crop in the ith command area, $/ha
Tsa simulation “temperature” during cooling schedule with dimensionless
Tnew,Told simulation “temperatures” at the end and beginning with dimensionless
W unit price of irrigation water, $/m3
x decoded decimal
Ya,season relative crop yield reduction due to infiltration over the entire season
Yam,season relative crop yield reduction due to water stress over the entire season
Yam,stage relative crop yield reduction due to water stress at each stage
Yi,j yields per hectare of jth crop in the ith command area, ton/ha
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