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Optimal Design of Irrigation Canal NetworkUnder Uncertainty Using Response SurfaceMethodAbdulmohsin Alshaikh a & Saud Taher aa College of Engineering King Saud University , P.O. Box 800, RIYADH, 11421,SAUDI ARABIAPublished online: 22 Jan 2009.

To cite this article: Abdulmohsin Alshaikh & Saud Taher (1995) Optimal Design of Irrigation CanalNetwork Under Uncertainty Using Response Surface Method, Water International, 20:3, 155-162, DOI:10.1080/02508069508686468

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Optimal Design of Irrigation Canal NetworkUnder Uncertainty Using Response SurfaceMethod

by Abdulmohsin Alshaikhand Saud TaherCollege of EngineeringKing Saud UniversityP.O. Box 800RIYADH 11421SAUDI ARABIA

ABSTRACT

Irrigation-water-delivery systems are designed and managed to receive water from a source and to distributeit among farmers in order to meet their agricultural requirements. High system performance can be achievedthrough rehabilitation of deteriorating and inadequate physical facilities and through improved system manage-ment. Various design decisions must be made in order to rehabilitate or develop irrigation-water-delivery systems,including those related to specification of the characteristics of hydraulic structures used to convey regulate, ordivert water

This study develops and applies a response surface methodology (RSM) for achieving optimal design forhydraulic structures in irrigation-water-delivery systems in canal networks. This approach provides a means ofunderstanding system behavior through developing a response surface in terms of a mathematical expressionrepresenting system performance as affected by design decisions. Design decisions include pipe diameters fordiversion and regulating structures. Simulation of steady spatially varied flow was incorporated into the responsesurface methodology to determine high-performance low-cost solutions.

Objectives of adequacy, efficiency dependability, and equity of water delivery were considered in defining waterdelivery performance. Fuzzy membership functions were used to address subjectivity associated with interpretingexpected values of performance measures associated with each of the prescribed objectives.

This study is an extension of a previous study by Alshaikh [l]. That study reported the application of RSMon a single canal case while herein RSM was used for the case of a canal network. Though, in general, RSMsfor large-scale branched systems are computationally intensive, this proposed methodology overcomes thisdrawback. The approach constitutes a significant easy-to-use step forward in the development of comprehensivesystems-scale techniques for the design of structural components of irrigation-water-delivery systems.

INTRODUCTION

Irrigation water delivery systems (IWDS) are designedand managed to receive water from a source and distributeit among farms where it is used to meet agriculturaldemands. The design or rehabilitation of these IWDSinvolves decisions about siting, dimensions, and charac-teristics of facilities for conveyance, regulation, monitor-ing, and diversion of flow. The characteristics of all theseindividual system components work together to determinethe performance of the entire IWDS. Therefore, an op-timal design of an individualan assessment of the entiremance.

hydraulic structure requiresdistribution system perfor-

Vagueness associated with definition of system objec-tives and interpretation of the associated performancemeasures (objective uncertainty) adds to the complexityof the problem. Objective uncertainty was addressed byGates and Alshaikh [4], and Gates et al. [5] in terms ofmethodology that is based on the concept of fuzzy setsand membership functions. In that study an optimaldesign criterion was formulated and a multiobjective

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Assessing system performance involves determining thevalue of performance measures. Molden and Gates [2]defined four measures for the objectives of adequacy,efficiency, dependability, and equity of water delivery inIWDS. These measures were used by Alshaikh [ 1], Heyderet al. [3], and Gates and Alshaikh [4].

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optimization model was developed and solved using bothRSM and Hooke and Jeeve pattern search technique. Thestudy also reported that response surface methods for thecase of a three-or-more-dimensional problem cannot bevisualized and a mathematical model representing theresponse surface must be determined.

The clear advantage of RSM is the useful informationobtained about the behavior of the response function overthe decision space. On the other hand, this method iscomputationally very intensive, unless the problem issimplified, especially for a nonlinear problem with manydecision variables.

The focus of this study was on the case of a branchedcanal network, which represents the most frequent sce-nario presented in irrigation delivery system design. Insuch a case, conditions in the farm laterals branchingfrom the distributary canal affect each other and shouldnot be designed or analyzed separately. So, the systemassumed here was designed as one unit including theinteraction between the different parts of the system. Forexample, if a large diameter pipe is designed for a certainfarm lateral, it may cause a decrease in water flow forthe other farm laterals. Also, hydraulic effects due to anyflow perturbation cause upstream propagation of theseeffects in canal networks with subcritical flow regimes.

In the present article, we extend the work by Gates etal. [5] by introducing a Regression Analysis-based meth-odology. Using this, the large-scale multidimensional com-plex problem will be simplified to the extent that it canbe handled efficiently by RSM. This approach will helpnot only in determining the optimum design or its vicinitybut also in studying the impact of a suggested hydraulicstructure design on the entire IWDS performance.

IRRIGATION DELIVERY TECHNICALAND ECONOMIC PERFORMANCE

Technical Performance

In assessing the performance of irrigation water deliverysystems, there are many measures that have been devel-oped in recent years. Among them, four measures asso-ciated with the objectives of adequacy, efficiency, depend-ability, and equity of water delivery, developed by Moldenand Gates [2], were selected to be used here. Thosemeasures are among the latest expressions of main irri-gation delivery system performance in which both timeand space frames are considered. These measures addressobjectives of technical performance. Economic perfor-mance was also considered by using a relative cost measurefor the designed structures.

An essential objective of water delivery systems is todeliver irrigation water in adequate amounts to meet farmneeds. Agricultural production, along with other irrigationproject goals, depends heavily on having adequate water.

The performance relative to adequacy, PA, for a given

irrigated region, R, served by the irrigation system overa time period, Z was proposed as:

pA = (+) F [(;) ;P”]in which

PA = ~~1~2~ i f QI, 5 QR= 1 otherwise,

QD = the actual delivered amount of water calculatedusing the Colorado State University Water De-livery Model (CSUWDM) discussed later, and

QR = the required (targeted) amount of water calcu-lated as follows:

QR =ET, x a x KC x 1000 x Y

E, x 1000 x 86400 x d x h

in whichQR = required flow of irrigation water at a given farm

turnout for a certain month, (l/s),ET, = potential crop evapotranspiration (mm/day),

a = area served by the farm turnout (m2),KC = crop coefficient for a given month (incorporates

the effect of crop growth stage, crop density, andother factors affecting ET),

Y = number of rotations to irrigate the system (inmany cases, the entire irrigation system is notirrigated at once; rather, it is irrigated in stages,or rotations),

E, = farm irrigation application efficiency,d = ratio of irrigation days to total days in the month

(days/days), andh = ratio of irrigation hours to total hours in a day

(hr/hr).

In this equation, PA is a spatial and temporal averageof the values of PA. Values of PA vary in space and timebecause of the variability of both QD and QR from timeto time and from one diversion point to another. Whenthe value of pa exceeds unity at a diversion point, that isQI, > QR, the case is considered fully adequate at thattime regardless of the amount of excess.

Delivering the irrigation water in an efficient way isfundamental to agricultural water conservation. Conser-vation is especially critical in regions suffering from watershortages. The measure of efficiency used here, PF, is:

in which

PF = QR/eD for QR 5 QD= 1 otherwise

Again, PF measures the system efficiency by taking thespatial and temporal average of the point measure ofefficiency, pF. When QD < QR, that is PF > 1, the valueof PF is taken as 1, which indicates that the case is fully

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efficient (no waste of water) regardless of the amount ofshortage in the delivered water.

Dependability of water delivery is associated with thetemporal variation in the ratio of the delivered amountof water to the required amount at diversion points inthe system. The less this variation, the more dependableis the system. Consistency in the delivered amount ofwater is important to farmers because it helps them toproperly plan their irrigation and farming operations. Thedependability measure, P,, is defined as:

P,= ;0

F CV, (QD/QR)

in which CV, (QJQJ is the temporal coefficient ofvariation (standard deviation divided by the mean) of theratio QD/eR over the time period under consideration.The system becomes more dependable as the value of PDapproaches zero.

Equity is associated with fairness of water distributionamong the users in the system. Having an unfair waterdelivery may cause many social problems among thefarmers such as fighting with each other. Such problemsmay even become the cause of a decrease in agriculturalproductivity. The performance measure relative to equity,PE1 is given as:

PE =0f F CV, (QdQd

in which CV, is the spatial coefficient of variation of theratio (QD/QR) over the region. The degree of equitybecomes greater as the value of PE approaches zero.

Table 1 summarizes the four performance measures forirrigation water delivery used in this study.

Economic Performance

In the design or rehabilitation of irrigation deliverysystems, the cost required to build the various systemcomponents is an important factor for the decision-makers. Minimizing the cost for the designed structures

Table 1. Summary of performance measures for Irrigation WaterDelivery Objectives [2]

System Objectives Performance Measures

Adequacy

Efficiency

Dependability

EquitypE= +, z ch@&R)

0 T

is considered in this study as an economic objective thatshould be met. This economic objective is addressed alongwith the previously mentioned technical objectives toachieve both low cost and high beneficial design for theselected hydraulic structures.

Regression analysis was performed on the gathered datafrom concrete companies in Fort Collins, Colorado, toobtain the following cost function:

C = 95.8 D’.47

in which

C = the cost of the reinforced concrete pipe ($/m oflength), and

D = the pipe diameter (m).

CHARACTERIZATION OF OBJECTIVEUNCERTAINTY

Irrigation water delivery systems are designed to meetcertain objectives as discussed before. Evaluation of howwell those objectives are met is accomplished using suit-able performance measures. The uncertainty associatedwith how to assess these performance measures in viewof system objectives is considered. This type of uncertaintyis called objective uncertainty and is associated with thevagueness and subjectivity associated with the interpre-tation of predicted system performance objectives.

The concept of fuzzy sets and membership functionsprovides a means to consider the uncertainty in assessingand judging the different system designs using values ofperformance measures. The set corresponding to theachievement of the performance objective can be modeledas a fuzzy set. This fuzzy set, A, can be defined ascomposition of those designs that achieve the objectiveconsidered Z. A membership function pZ maps valuescontained in a collection of values that may be taken bythe values (PJ of a performance measure estimated for agiven design and associated with the performance objec-tive Z. Those mapped values indicate the degree ofmembership in which a certain strategy is represented inthe fuzzy set A,. A value of pZ = 1 indicates completemembership in A, and P, = 0 indicates nonmembership[6]. Values between zero and one indicate correspondingrelative degrees of membership. An example of a mem-bership function is one that was applied to the Alamosa-La Jara irrigation-water-delivery systems in the San LuisValley, U.S.A. for the objectives of adequacy, efficiency,dependability, and equity [ 3].

An additional benefit of using membership functionsfor this study was that they map the values of performancemeasures into the interval [0, 1], which solves the problemof noncommensurate and unbounded measures that canarise when using multicriterion objective functions forthe optimization process, as was the case for this presentstudy. Different methods for constructing membershipfunctions are suggested in Norwich and Turksen [7].

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Figure 1 shows the membership functions used for thisstudy. More explanations about those functions are avail-able in Alshaikh [ 1].

OBJECTIVE FUNCTION FORMULATION

Now that the system technical performance measures,the economic performance measure, and the membershipfunctions have been defined, the next step is to formulatethe objective function that represents the overall systemperformance (IWDSP).

The mathematical formulation consists of consideringa weighted criterion function, Z, of a technical perfor-mance measure, P, and a relative cost, C, associated withthe design variables, D. The design vector, D, includesthe design characteristics of the selected structures suchas pipe diameters for submerged orifice diversion struc-tures. Thus, the form of the objective function is:

z = W,P - W,C

subject to:

(1) the system hydraulic equations and boundary con-ditions solved by CSUWDM discussed in the fol-lowing section

and

(2) kin 5 D 5 Max

in which P = weighted average of the membership func-tions Pi, Pi, pD, and pE associated with the values ofperformance measures of adequacy, efficiency, depend-ability, and equity respectively:

1 .o

P* 0.5

E

a.

00 0.5 1.0

1.0

PLD 0.5

0

1 . 0

!-$ 0.5

L#Izi!l

b.00 0.5 1.0

PF

1 . 0 c-\PE 0.5L!Lld.

0

Figure 1. Membership functions for objectives of (a) adequacy; (b)efficiency; (c) dependability; and (d) equity of water delivery for examplecanal network.

158

p = VA PA + WF PF + WD PD + WE PE

and in which

WA = relative weight associated with the objective ofadequacy,

WF = relative weight associated with the objective ofefficiency,

WD = relative weight associated with the objective ofdependability,

WE = relative weight associated with the objective ofequity, with z (WA + WF + W, + WE) = 1,

W, = relative weight associated with technical perfor-mance,

W, = relative weight associated with relative cost with(w, + w,) = 1,

C, = relative cost associated with the design D =C(DK’(DmJ,

C = cost associated with a given design D,Dmax = upper bound value of D, andDmin = lower bound value of D.

Having the different weights included in the objectivefunction gives the decision-makers the flexibility to rankthe different objectives according to their priorities.

For calculating Z, system hydraulic equations andboundary conditions are solved by CSUWDM (or anyother suitable model) as discussed in the following section.

PHYSICALLY BASED WATER DELIVERYSIMULATION MODEL

The CSUWDM computer model was used to simulateflow in the canal network systems considered in this study.The model simulates steady, spatially-varied flow in anetwork of nonprismatic open channels. Flow throughsubmerged orifice turnouts (gated or ungated), siphonturnouts, submerged culverts, weirs, flumes, check struc-tures, bends, expansions, and contractions can be analyzed[ 8 ] . Flow depth, average velocity, and flow rates at selectedlocations within the system are computed. The majorinputs to the model are the water delivery schedule,boundary conditions, and hydraulic parameters for thechannels and structures. For more explanations about themodel, see Gates et al. [8].

SOLUTION SCHEME USING RESPONSESURFACE TECHNIQUES

Response surface methodology is a set of techniquesthat encompass setting up a series of experiments thatwill yield adequate and reliable measurements of a re-sponse function of interest and developing a mathematicalmodel that allows a determination of the maximum (orminimum) values of the response [9].

To find the optimum or its vicinity, sampling or searchtechniques and regression analysis may be used [9]. Re-

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sponse surfaces for N2-dimensional decision spaces, can-not be visualized. A mathematical model must be builtin which the response surface is expressed as a functionof the input variables in the form of a polynomial orother suitable expression. A suitable optimization tech-nique can then be used to find the solution for the optimalvalue.

The clear advantage of using this technique is the usefulinformation obtained about the behavior of Z over thedecision space. On the other hand, the method is com-putationally very intensive, unless simplified, especiallyfor nonlinear problems with large numbers of decisionvariables. The solution scheme is designed in the followingsteps:(1)

(2)

(3)

(4)

Define the studied IWDS and its structures. Thatmeans to collect data about the physical componentsof the system including its hydraulic structures’ di-mensions.Find those structures where IWDSP is sensitive to.This is done by skeletonizing the complex system inorder to find the structures that have greater effect,when their dimensions change, on the system behav-ior. Those structures are called the critical structures.Develop a mathematical model to relate IWDSP tothe diameter size of the critical structures found fromstep 2. A suitable mathematical expression can beassumed and regression analysis can then be used toestimate the model parameters.Find the optimal design, using the equation derivedfrom step 3. A suitable optimization method can beused to get the optimal set of structures dimensionsin which IWDSP is maximized. Sensitivity analysiscan also be performed.

Hypothetical Site and Irrigation Schedule Description

The hypothetical system assumed for the present studywas a distributary canal, 4,100 m in length, taking waterfrom a nearby stream and delivering it to ten farm lateralsthrough submerged pipe diversion structures located atvarious stations along the distributary canal. These farmlaterals in turn delivered water to a total of 108 farmturnouts spaced every 50 m along the farm laterals. Theturnouts to the farm laterals were assumed to be gatedbut unregulated and the farm turnouts were assumed tobe ungated. This arrangement is typical in many parts ofthe world where there is a desire to have less operationaland maintenance responsibility and lower costs. The entiresystem was assumed to be lined so that the geometry ofthe canal and laterals and the bed slope do not varyconsiderably. Figure 2 shows the layout of the hypotheticalsystem and Table 2 summarizes the different character-istics of the system.

The design decision variables in this example were theten diameters of the pipe turnout structures that divertedwater under submerged flow conditions from the distri-butary canal to the farm laterals.

Farm Lateral 1:6 Turnouts

4Farm Lateral 2:12Turnouts

wFarm Lateral 3:6 Turnouts

IFarm Lateral 4:15 Turnouts

IFarm Lateral 5:10 Turnouts

iFarm Lateral 6:12 Turnouts

1Farm Lateral 7:10 Turnouts

IFarm Lateral 8:15 Turnouts

IFarm Lateral 9:12 Turnouts

1_ Farm Lateral 10:

10 Turnouts

Figure 2. System layout used for branched canal network.

The area served by the system could be calculated sincethe layout of the entire system was fixed. Areas servedby every farm lateral and then by every farm turnoutalong that lateral are shown in the last two columns ofTable 2. The total area served by the hypothetical systemwas 232.61 ha. The largest area served by a farm lateralwas that served by farm lateral number 4 which was 38.22ha, and the smallest one was that of farm lateral number1 which was 7.52 ha. The areas served by the farm

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turnouts were used to calculate the required amount ofwater, QR, for each farm turnout for each month of theirrigation season as discussed in a previous section.

For water delivery scheduling, the system was assumedto be operated under a fixed rotation schedule with tworotations. The first rotation served the five upstream farmlaterals; the second served the five downstream farmlaterals. Water was assumed to be delivered to the distrib-utary canal for seven days and then shut off for sevendays. The irrigation period was twelve hours daily. Theassumed crop was cotton planted in the summer monthsfrom April to October. According to Doorenbos and Pruitt[10], the monthly crop factor, K, for cotton from Aprilthrough September for arid areas is: 0.35,0.50,0.90, 1.05,1.0, and 0.75, respectively. This water delivery schedulewas used to calculate the values of the water requirementsat each of the farm turnouts.

Values of evapotranspiration (ET), water surface level(WSL), application efficiency (E,), and Manning rough-ness coefficient (n) are given by Alshaikh [l] dependingon data derived from Abysha canal in Egypt. Table 3shows the statistics of data used.

CASE ANALYSIS

Skeletonizing the studied complex system requires find-ing the critical turnouts. This was accomplished by firstassigning three levels of diameters (0.2 m, 0.35 m, and0.5 m) for each of the ten turnouts along the network.Second, evaluating the IWDSP for each of these three

design levels for every turnout taking one at a time whilekeeping the other turnout diameters constant at theminimum level of 0.2 m. Finally, selecting those turnoutspossessing the highest coefficient of variation, CV, reflect-ing largest impact on IWDSP As shown in Table 4,critical turnouts are numbers 1, 2, 4, and 6 where CV is2 0.05.

After determining the critical turnouts, (N=4) a 4-dimensional mathematical model was developed. In de-veloping this model, a second-order polynomial expres-sion was assumed for its popularity. However, the poly-nomial model gave a high correlation coefficient, R2 asshown later. The mathematical model is represented asfollows:

Z (or IWDSP) = a + i bi Di + 5 Cj Dfi = l i - l

n-l m

+ 2 2 du Di Dji=l j - 2

in which D1, Dz . . . . D, are the input variables (the turnoutdiameters here) that influence the response Z (IWDSPhere), a, b, c, and di,. are unknown parameters that needto be estimated.

A design by means of which values of the response arecollected for estimating the parameters in the abovesecond-order model is called a second-order design [9].A second-order design developed by Box and Behnkenwas used here. Twenty-five points of different combina-tions of D1, D2, D4, and D, were collected. The corre-sponding values of Z (IWDSP) were calculated using

Table 2. Characteristics of the hypothetical branched canal network system

Locationof EL. along Bottom Farm

Length Distributary Bed Slope Side Slope Width No. of Total Area TurnoutCanal (m) (m) (m/m) (H:V) (m) Turnouts Served (ha) Area (ha)

Distributary 4,100 0.001 0 10 232.61 -EL.1 350 Jo 0.002 1:2

;36 1.52 1.25

EL.2 650 400 0.001 1:2 0:35 12 32.60 2.72EL.3 350 1,100 0.001 1:2 0.35 6 16.71 2.78EL.4 800 1,400 0.002 1:2 0.35 15 38.22 2.55EL.5 550 2,100 0.001 1:2 0.35 10 31.54 3.15EL.6 650 2,600 0.003 1:2 0.35 12 21.72 1.81EL.7 550 2,800 0.002 1:2EL.8 800 3,200 0.001 1:2

8.:5 10 15.77 1.5815 30.58 2.04

EL.9 650 3,600 0.003 1:2 0.35 12 24.85 2.07EL.10 550 4,000 0.002 1:2 0.3 10 13.13 1.31

EL. = Farm Lateral

Table 3. Statistical characteristics of the used parameters in CSUWDM for the example canal network

WSL (m above Mean Sea Level) ETo (mm/day)

Statistics April May June July August Sept. April May June July August Sept. E, n

Probability normal normal normal normal normal normal normal normal normal normal normal normal truncated truncateddensity normal logkxtion normal

Mean 41.12 41.14 41.31 41.43 41.40 41.16 8.2 9.0 11.3 11.4 10.4 9.3 0.74 0.015

Standarddeviation 0.21 0.21 0.16 0.11 0.11 0.11 0.14 0.41 0.20 0.15 0.22 0.40 0.22 0.004

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Table 4. Coefficient of variation for turnouts Table 5. The combination of critical turnouts diameters and thecorresponding IWDSP values

D1 DZ D4 D6 ZTurnouts CV

1 0.062 0.053 0.03

0.200.500.200.50

0.200.200.500.50

0.350.350.350.35

0.350.350.350.350.200.200.500.50

0.3590.3290.3820.3430.3290.3430.3270.3540.3630.3650.3230.3480.3250.3340.3470.3500.3830.3400.3400.3280.3510.3570.3310.3890.346

4 0.06

ii0.030.05 0.35

0.350.350.35

0.200.50; 0.04

0.049 0.04

10 0.04

0.35 0.350.350.350.350.350.350.350.20

0.200.500.350.350.35

8 0.359 0.35 0.35

0.2010 0.200.500.200.500.35

CSUWDM. The different combinations of diameters andthe corresponding Z values are shown in Table 5.

1112

0.200.500.500.350.350.350.350.20

0.350.3513

14

Regression analysis was then applied to find the bestfit model and to estimate the unknown parameters of theassumed polynomial equation. In the following, the valuesof these parameters and their correlation coefficient, R2,are given:

a = 0.220; bq = 0.463; cq = -0.896; d,, = 0.222b, = 0.013; cl = -0.141; d,z = -0.100; dz4 = 0.122b2 = 0.168; cz = -0.269; d,, = -0.189; d34 = 0.144b, = 0.141; c3 = -0.180; d,d = 0.211; R* = 0.98

In order to verify the accuracy of the developed modelin representing the actual response of IWDSP, a compar-ative analysis was performed. Many design combinationsof all system turnouts, excluding the ones used in modeldevelopment, were randomly chosen. These chosen com-binations were used to calculate IWDSP (Z) by first usingthe developed polynomial model and then by using thepreviously mentioned objective function. Results fromboth models were then compared showing a relativelyreasonable difference of 11 per cent.

The above developed model can then be used toestimate IWDSP at any given combination of turnoutdiameters. Sensitivity analysis was applied to evaluatechanges in IWDSP resulted due to changes in turnoutdiameters. For D, as an example, the values of 0.2 m,0.35 m, and 0.5 m were selected and the correspondingIWDSP values were estimated using the developed modeland a plot is drawn to show this result. The same thingwas drawn for the other three diameters (D2, Dq, and 06).Figure 3 shows these plots. IWDSP was found to be moresensitive to D, than other critical diameters.

It can be stated, from the above figure, that no generaltrend between IWDSP and D was realized. This is referredto the complexity of such a relation. In order to find theoptimal solution in which IWDSP is maximized, theHooke and Jeeves pattern search technique was used.This optimization method is a very widely used searchmethod because it manages attempts in a simple thoughingenious way to find the most profitable search direction[ 1 I]. Descriptions of this method are given by Hookeand Jeeves [ 12]. When using this method, values of D,= 0.2 m, D, = 0.35 m, D, = 0.5 m, and D, = 0.35 mwere the results found to comprise the optimal set andIWDSP was equal to 0.397.

0.200.200.500.500.350.350.350.35

15 0.35 0.5016 0.35 0.2017 0.35 0.35

0.200.500.200.50

18 0.3519 0.35 0.20

0.5020 0.350.350.200.500.200.50

2122

0.500.350.350.350.35

0.350.35

0.200.2023

2425

0.350.35

0.500.50

SUMMARY AND CONCLUSION

An approach to finding the optimal design of hydraulicstructures in complex IWDS was presented. Four perfor-mance measures on IWDSP objectives (adequacy, effi-ciency, dependability, and equity) were used to evaluatethe I W D S P in conjunction with economical considerationof the designed structures. Fuzzy membership functionswere applied for these measures. The presented approachdepends on selecting the most critical structures on IWDSP,and developing a mathematical model. This developedmodel is used then to find the optimal design throughapplying a suitable optimization technique. Applicationof the approach to a hypothetical case study was discussed.

The developed model for the case studied here showedthat variation of IWDSP to changes in turnout diameterswas not significant (ZWDSP range was 0.304 to 0.389,i.e., 28 per cent change). This is referred, in part, to thefact that only four diameters among the 10 availablediameters were considered in the developed model. Thelarger the number of diameters considered, the morecomplex the model will be. The degree of accuracy of theresults will be influenced by this simplification.

The optimal set of diameters for the turnouts wasestimated using the Hooke and Jeeves optimizationmethod. This set is at 0.2 m, 0.35 m, 0.5 m, and 0.35 mfor the critical structures D1, D2, D4, and D6 respectively.The complexity of the problem of irrigation networksrequires that we simplify it. This simplification, in turn,leads to an inaccurate result. This is one important reasonnot to presume that this estimated optimal set is theglobal optimum. But, at least this obtained solution fitsin the vicinity of optimality.

Nevertheless, the importance of the suggested approach

Vol. 20, No. 3 (1995) 161

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0.32

0.346

0.342

0.338

0.34

0.33

0.32

/

I ’ I ’ I ’

/-

c. D,I I I I I I

0.2 0.3 0.4 7 0.5METERS

Figure 3. Response of IWDSP to turnout diameter.

0.6

is not limited or downgraded by results from a certaincase study. Large branched irrigation canal systems areusually hydraulically complex. The presented approachhelps to deal with this complexity; this is, in turn, valuableto complex irrigation project managers.

Finally, the approach herein is not limited to certaindesign features of IWDS components or certain mathe-matical models. Optimal design of canal slope and width,and other characteristics of IWDS can be estimated to areasonable degree of accuracy.

REFERENCES

1. Alshaikh, A., “Stochastic Optimal Design of HydraulicStructures in Irrigation Water Delivery Systems,” Ph.D.Dissertation, Colorado State University, Fort Collins,Colorado, U.S.A., 1992.

2. Molden, D., and T. Gates, “Performance Measures forEvaluation of Irrigation-Water-Delivery Systems,” Jour-nal of Irrigation and Drainage Engineering, ASCE, Vol.116, No. 6, 1990, pp. 804-823.

3. Heyder, W., T. Gates, D. Fontane, and D. Salas, “Multi-criterion Strategic Planning for Improved Irrigation De-livery II: Application,” Journal of Irrigation and Drain-age Engineering, ASCE, Vol. 117, No. 6, 1991, pp. 914-934.

4. Gates, T., and A. Alshaikh, “Stochastic Design of HydraulicStructure in Irrigation Canal Networks,” Journal of Ir-rigation and Drainage Engineering, ASCE, Vol. 119, No.2, 1993, pp. 346-363.

5. Gates, T., A. Alshaikh, S. Ahmed, and D. Molden, “Op-timal Irrigation Delivery System Design under Uncer-tainty,” Journal of Irrigation and Drainage Engineering,ASCE, Vol. 118, No. 3, 1992, pp. 433-449.

6. Zimmermann, H., Fuzzy Sets, Decision Making and ExpertSystems, Kluwer Academic Publishers, Boston, Mass.,U.S.A., 1987.

7. Norwich, A., and I. Turksen, “A Model for the Measure-ment of Membership and the Consequences of Its Em-pirical Implementation,” Fuzzy Sets and Systems, Vol.12, No. 1, 1984, pp. l-25.

8. Gates, T., D. Molden, W. Ree, M. Helal, and A. Nasr,“Hydraulic Design Model for Canal Systems,” Proceed-ings, 2nd Conference on Microcomputers in Civil Engi-neering, Univ. of Central Florida, Orlando, Florida,U.S.A., 1984.

9. Khuri, A., and A. Cornell, Response Surfaces: Design andAnalysis, Marcel Dekker, New York, N.Y., U.S.A., 1987.

10. Doorenbos, J., and H. Pruitt, “Guidelines for PredictingCrop Water Requirements,” Irrigation and DrainagePaper No. 24, EA.O., Rome, Italy, 1977.

11. Walsh, G.R., Methods of Optimization, John Wiley andSons, New York, N.Y., U.S.A., 1975.

12. Hooke, R., and T. Jeeves, “Direct Search Solution forNumerical and Statistical Problems,” Journal of Com-puter Machinery, Vol. 8, No. 2, 1961, pp. 212-229.

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