wische: a dss for water irrigation scheduling
TRANSCRIPT
ARTICLE IN PRESS
Omega 38 (2010) 492–500
Contents lists available at ScienceDirect
Omega
0305-04
doi:10.1
$This
Educati
of Scien� Corr
E-m
(L.F. Esc
a.rabasa
journal homepage: www.elsevier.com/locate/omega
WISCHE: A DSS for water irrigation scheduling$
M. Alminana a, L.F. Escudero b, M. Landete a, J.F. Monge a,�, A. Rabasa a, J. Sanchez-Soriano a
a Centro de Investigacion Operativa, Universidad Miguel Hernandez, Elche (Alicante), Spainb Departamento de Estadıstica e Investigacion Operativa, Universidad Rey Juan Carlos, Madrid, Spain
a r t i c l e i n f o
Article history:
Received 13 January 2009
Accepted 22 December 2009
Processed by B. Levdaily irrigation for a given land area by taking into account the irrigation network topology, the water
volume technical conditions and the logistical operations. The system has been validated by the
Available online 4 January 2010Keywords:
Water resource scheduling
Agricultural irrigation
Mixed 0–1 separable nonlinear problem
83/$ - see front matter & 2009 Elsevier Ltd. A
016/j.omega.2009.12.006
research has been partially supported by th
on and FEDER funds through the Grant MTM2
ce and Innovation through the Grant MTM20
espondence author.
ail addresses: [email protected] (M. Alminana), la
udero), [email protected] (M. Landete), monge
@umh.es (A. Rabasa), [email protected] (J. San
a b s t r a c t
In this paper we present the models and the algorithms which are being used in a decision support
system (DSS) to determine water irrigation scheduling. The DSS provides dynamic scheduling of the
Agriculture Community of Elche (Spain) and annexed to their Supervisory Control and Data Acquisition
system (SCADA). We present two heuristic approaches to solve the mixed 0–1 separable nonlinear
program for irrigation scheduling implemented with free software.
& 2009 Elsevier Ltd. All rights reserved.
1. Introduction
In 1968 the Council of Europe published the European WaterCharter (http://assembly.coe.int/), which states fundamentalprinciples for the conservation of water resources and establishescriteria for their rational use. Besides outlining the fundamentalprinciples for the protection of this indispensable and vital asset,the European Water Charter points out the need for the inventory,control and management of water resources.
The need for rational water management has become greaterin many Mediterranean regions as a result of changes in theavailability of water, changes in general climatic conditions andthe adverse effect of the actions of human beings on theenvironment. See [6] and the references therein for more detailsof water-related problems in Mediterranean regions.
Over the last years the traditional irrigation schedulingsystems have been changed for other new systems in thesoutheast of Spain because of the scarcity of water resources inthis arid region. The traditional inundating irrigation systemshave been progressively substituted by drop (sprinkler) irrigationsystems, in which the water is channeled to the irrigation pointswhere it is necessary and the required quantity is completelyregulated and controlled. This kind of irrigation systems is morecommon day by day in the above-mentioned region.
ll rights reserved.
e grants from the Ministry of
004-01095 and the Ministry
09-14087-C04-01, Spain.
@umh.es (J.F. Monge),
chez-Soriano).
The ability to help the decision maker in the planning andscheduling of the distribution of water resources depends on thelevel of sophistication of the tools and techniques available. Acomprehensive approach may remedy the inadequacies of thetools currently available, by developing a hydrologic modelingframework and a highly numeric intensive computation decisionsupport system. See [14–17,20] and the references therein.
We should differentiate between water resource planning overa time horizon which is usually long and water distributionscheduling which is usually on a daily basis. For the case ofplanning see [3] for the deterministic environment, and [13,18]for the stochastic case by considering the uncertainty in the mainparameters (water inflow and needs).
The Agriculture Community of Elche (ACE), Elche being a cityin the southeast of Spain, is constantly looking for new irrigationsystems in order to conserve the natural environment and to saveas much water as possible, because they realize the future of theregion depends in some way on the management of the scarceand important resource which water is. At this moment, ACE is ina first phase of modernization of the channeling systems and thedistribution of irrigation water among its members. The firstphase in this work was the substitution of the water canalizationfor a new system consisting of underground pipes and pumps. Thepumps send the water from the dam to each irrigation area. Thesystem is controlled by means of a SCADA (Supervisory Controland Data Acquisition) which controls and regulates the availableflow in each irrigation area. The advantages of this improvementto the installations are obvious. On the one hand, the water lost byevaporation and filtering is reduced and, on the other hand, eachmember of ACE has the guarantee of a fixed quantity of irrigationwater with a specific pressure on his land.
This improvement in the infrastructures of the irrigationsystem is very important in order to save irrigation water but
ARTICLE IN PRESS
UserAdministrator
WISCHE SCADA
inputdata
reportstelemetry
control
Fig. 1. Components of the decision support system and flow of data.
M. Alminana et al. / Omega 38 (2010) 492–500 493
there are two technical problems to be resolved: (i) the design ofthe pipe network does not guarantee the irrigation service to allmembers of ACE simultaneously, therefore some priority criteriaare needed in the management of the system; (ii) the overallpressure of the pipe network has to be controlled to avoidpossible breakage of the pipes or water losses. To determine theirrigation scheduling we have considered a set of time periods.The scheduling interval is five periods per day of 4 h each, thenumber of daily periods can be changed by system requirements.In each period the members of ACE are divided into two groups:active and non-active. Each member of the active group has acertain water volume and a minimum pressure in each period oftime guaranteed, and each member of the non-active group hasthe service of the irrigation water blocked. The SCADA controlsthe opening and closing of the valves.
The DSS WISCHE (Water Irrigation for SCHEduling) provides asolution to the problem of assigning each member of ACE to a setof consecutive time periods in the daily irrigation scheduling,such that the water volume and the minimum service pressureare guaranteed. In addition, the solution provided guarantees thatthe water speed in the network does not exceed a previously fixedmaximum value.
As the preferences of the members of ACE over all daily periodscould be the same or coincide in some periods and, very likely, itis impossible to satisfy all preferences simultaneously, a specialmodule has been incorporated into the system which records thepast consumption of water and records the scheduling forprevious time periods for each hydrant. Thus, we are able todetermine indices of inefficient use and the time period assignedfor each hydrant in previous days. These indices allow us todetermine an irrigation earliness-tardiness unit cost for eachmember of ACE. The optimization model implemented in thesoftware minimizes the daily irrigation earliness–tardiness cost ofthe users. It is a mixed 0–1 separable nonlinear model presentedin [1], that for completeness we include below.
For a good exposition of mixed 0–1 linear programming, seee.g. [22], and see [4] for a mixed-integer linear programmingmodel in a irrigation scheduling problem in another context.A linear integer programming for scheduling decision at adistributor of industrial gasses is presented in [9].
The decision support system presented in this work has beentested by solving a real-life problem presented by ‘‘La Comunidadde Regantes, Riegos de Levante, Canal 2nd’’, which belong to ACE.Its irrigation area comprises 2188 Ha and is distributed in 20 pipesectors (i.e., 20 head nodes) with a total number of 2831 nodes(2025 of them are hydrants with their own water demand needs).The irrigation is needed on a daily basis for a set of time periods.The water flows from a reservoir with a capacity of 13 Hm3, andthe full system has an arborescent structure.
The remainder of the paper is organized as follows. In Section 2we describe the structure of the decision support system WISCHE.Section 3 presents the irrigation earliness–tardiness unit costsystem for the hydrants. Section 4 introduces de mixed 0–1separable nonlinear model. In Section 5 we present the optimiza-tion algorithms implemented in the software WISCHE, as well asthe results of the computational experiment that has been carriedout for validating the system. Finally, Section 6 concludes.
2. WISCHE structure
WISCHE is a decision support system annexed to a SCADA. Itsends to the control system the daily scheduling of the irrigationsystem, providing information about which hydrants will beserved in each time period, and receives information from thecontrol system on the water consumption of each hydrant. A
scheme illustrating the implemented systems and the relationbetween them is shown in Fig. 1. WISCHE consists of fourmodules. The first module processes historical data of telemetry,debugs reading errors and provides tools to create graphs of waterconsumption, pressure, etc. The second module allows tointroduce the irrigation target starting time period for eachhydrant user. Additionally it allows to modify the priority criteria.The third module is responsible for processing all the informationin the file telemetry, together with the history of the memberspreferences and their new preferences for the following day orweek. Taking into account all this information, the modulegenerates a set of irrigation earliness–tardiness unit costs foreach hydrant and time period. This set of priorities is used by theoptimization module that provides the assignment of irrigationperiods to each member of ACE. The diagram shown in Fig. 2presents the structure of the files that the WISCHE and SCADAmodules share.
3. Earliness–tardiness cost for irrigation scheduling
It is probably impossible to satisfy the preferences of all themembers of ACE because of the design and dimension ofthe irrigation network and the constraints on the pressure andthe speed of the water. So, it is absolutely necessary to have amechanism which penalizes the assignment of a particularhydrant to an irrigation time period when there are severalhydrants with the same or coincident preferences.
The WISCHE system receives the irrigation target time periodsof the members of the ACE in a table-form. These preferences arethen combined with the past history of use of the hydrant. Wehave taken into account two significant factors to determine theassignment of a hydrant to a particular irrigation period:
�
Factor 1 ðF1htÞ: Inefficiency in the use of the assigned timeperiods starting at time period t in a number of previous dailyschedules. It is measured as the fraction of the number ofprevious days that hydrant h has been assigned to a set ofconsecutive time periods starting at time period t and notusing these time periods.
� Factor 2 ðF2htÞ: Fraction of the number of irrigation targetperiods starting at time period t that in fact have been assignedto hydrant h in previous daily schedules.
Both factors provide two indices (in (0,1)) which measure, foreach hydrant, the inefficiency in the use of the target irrigationperiod (Factor 1) and the fraction of times that their target periodshave been assigned (Factor 2). These factors are used by thesystem to calculate the hydrants’ weights in the assignment of theirrigation scheduling, i.e., the weight wht is evaluated as aweighted average of both factors, such that wht ¼ aF1
htþð1�aÞF2ht
with 0rar1. It is worthy to note that the weights wht can bedifferent for each hydrant h along the time horizon since thepreferences of a hydrant are not homogeneous along the time
ARTICLE IN PRESS
WISCHE
historyrequest
file
TELEMETRY
REQUEST
SCHEDULING
telemetry file
topology file
request file
SCADA
Fig. 2. WISCHE elements.
M. Alminana et al. / Omega 38 (2010) 492–500494
horizon. Notice that if a hydrant request to start the irrigation at agiven time period and, in the recent past, its irrigation has beenassigned to the requested target time period but it has not madegood use of the time interval for irrigation, then, it will have ahigh value in both factors and then a high coefficient wht . See[10,11] for other suggestions on how the weight wht can beestimated. In [2] a DEA model to estimate the weight ofagricultural agents is presented, the DEA methodology can helpus to evaluate the weight (efficiency) for each hydrant about theprevious consumption of water. However, in our system theearliness–tardiness unit cost of hydrant h for the irrigationstarting time period t is computed as follows:
cht ¼whtjt�thj ð1Þ
where th is the target irrigation starting time period for hydrant h. Ifthe hydrant’s scheduled irrigation starts early (i.e., toth) it is calledan early hydrant and if it starts late (i.e., t4th) it is called a tardyhydrant. In [10–12] an analogy between the irrigation schedulingproblem and the multi-machine job scheduling problem isestablished, so the problem is NP-hard. For excellent overviews ofjust in time schedules in OR, see [5,7,19], among others.
4. Mathematical models for irrigation scheduling
In order to describe mathematically the irrigation schedulingproblem, we need to take into account technical, topographicaland logistic irrigation scheduling parameters. The technical andtopographical parameters are related to the physical properties ofthe fluids. The logistic parameters are related to the managementof the irrigation system. Finally, the decision variables are relatedto the water volume through each node at each time period and towhen each hydrant may sink down.
The notation for all relevant elements and parameters in theWISCHE system is the following:
Sets of general elements:
T
set of time periods for water irrigation purposes. H set of hydrants in the geographical area under consideration. H0 set of bifurcation nodes.Ht �H
subset of hydrants whose irrigation starting period hasalready been fixed to time period t, tAT .Rh
set of upstream nodes to hydrant h in its path back to itssector head, including itself, for hAH [H0.Sh
set of successor nodes to node h, including itself, forhAH [H0. Notice that the nodes in set Sh can belong todifferent successor paths (this is the case where thesuccessor path has bifurcation nodes).T th
set of consecutive time periods of the irrigation byhydrant h if the hydrant h starts the irrigation at time t,for hAH, tAT . Notice that if the duration of irrigation in
hydrant h is, say, nh time periods, the irrigation is carriedout in the periods from set Tt
h ¼ ft; . . . ; tþnh�1g.
G set of sector heads, and gðhÞAG is the root node (sectorhead) of the subtree to which hydrant h belongs.
Technical and topographical parameters:
Eh
elevation of node h, for hAH [H0 [G. Dh diameter of the immediate upstream pipe segment of node hfor hAH [H0 [ G.
K hydromodule (l/s/ha), i.e., constant to obtain the watervolume to irrigate the land area through any hydrant at anytime period.
Lh
length of the immediate upstream pipe segment of node h,for hAH [H0 [ G.Pg
water pressure at sector head g, for gAG. Pmin minimum water pressure required by any hydrant at anytime period.
fht friction factor for obtaining the pressure in the immediateupstream pipe segment of hydrant h at time period t, forhAH [H0; tAT .
g
gravity acceleration coefficient. vmax maximum water speed allowed along the immediate up-stream pipe segment of any hydrant or bifurcation node atany time period.
Logistic parameters provided by the system operator:
Fh
effective land area (ha) to be irrigated by hydrant h, for hAH. cht earliness–tardiness unit cost for selecting hydrant h to start anon-preempted irrigation at time period t, for hAH:
yht fixed value to 0 or 1 for the variable yht due to logisticconsiderations, for hAHt . It allows the user to force thehydrant h to start ate time period t.
Variables:
qht
water volume through hydrant, bifurcation node or sectorhead h at time period t to satisfy its own needs, if any, and thewater needs of its successor nodes, for tAT ;hAH [H0 [ G.yht
0–1 variable, such that its value is 1 if the irrigation in hydranth start at time period t and, otherwise, it is zero, fortAT ;hAH.4.1. M_ETC: minimization of the earliness–tardiness cost for
water irrigation
The mathematical expression of the mixed 0–1 separablenonlinear model for minimizing the earliness–tardiness cost in
ARTICLE IN PRESS
M. Alminana et al. / Omega 38 (2010) 492–500 495
the irrigation planning and scheduling problem is as follows:
minXhAH
XtAT
chtyht ð2Þ
s:t: PgðhÞ þEgðhÞ�Eh �X
jARh
8fjtLj
p2gD5j
q2jt
!ZPmin 8tAT ; hAH
ð3Þ
qht ¼X
h0ASh�H0�G
KFh0
Xt0AT:tATt0
h
yh0t0 Þ 8tAT ; hAH [H0 [G
0@ ð4Þ
4
pD2h
!qht rvmax 8tAT ; hAH [H0 [G ð5Þ
XtAT
yht ¼ 1 8hAH ð6Þ
yht ¼ yht 8hAHt ; tAT ð7Þ
yht Af0;1g 8tAT ; hAH ð8Þ
Constraint (3) forces that the water pressure at any hydrantmust not fall below the specific threshold. The water volumethrough each node during each time period is defined byconstraint (4). The water speed allowed along the immediateupstream pipe segment of any node is enforced by constraint (5).Constraint (6) forces hydrant h can only be turned on once, say attime period t0. It is kept open during the time periods in T t0
h .Constraint (7) fixes the irrigation schedule for some hydrants, dueto logistics considerations imposed by the system operator.
4.2. MM_S: minimization of the maximum water speed
The model to minimize the maximum water speed is asfollows:
min vmax ð9Þ
s:t: ð3Þ2ð4Þ; ð6Þ2ð8Þ ð10Þ
4
pD2h
!qht rvmax 8tAT ; hAH [H0 [G ð11Þ
where vmax is the variable that gives the maximum water speedalong the time horizon.
Table 1Model dimensions for the pilot case.
m 44 390
nc 14 155
n01 9848
nel 332 231
dens 0.031 %
4.3. MM_P: minimization of the maximum water pressure
Another interesting objective function is the minimization ofthe maximum water pressure in the irrigation network at anytime period.
min Pmax ð12Þ
s:t: ð3Þ2ð8Þ ð13Þ
PgðhÞ þEgðhÞ�Eh�X
jARh
8f jtLj
p2gD5j
Q2jt
!rPmax; 8hAH [H0; tAT
ð14Þ
where Pmax is the variable that gives the maximum water pressurealong the time horizon.
5. Optimization algorithms
The optimization module needs four parameters, namely,number of irrigation time periods, maximum speed of the water,minimum pressure and pressure in the head hydrants.
The two approaches that we propose for solving the modelspresented in the previous section are as follows:
�
LAZY algorithm a l�a Guignard: An optimization approach byusing the free GLPK library: Approximated procedure to solvethe mixed 0–1 separable nonlinear model for irrigationscheduling. � HEUR: A heuristic algorithm to find a feasible solution in arelatively short computing time.
Depending on the characteristics of the instance, M_ETC can benon-feasible. In this case, the modification of certain constraintsin the system could be required, either the minimum pressure orthe maximum speed of the water. The user can compute thesmallest maximum speed of the system with all other parametersfixed, so he can adapt the maximum speed parameter to obtain afeasible value for the instance on hand. For example, the defaultmaximum value in the system for the pilot case is 2.5 m/s. Thismaximum speed is feasible in the whole system, provided theirrigation schedule for the 2831 hydrants is in five periods perday, 4 h per period. If the user of the control system makes thedecision to offer only three irrigation periods per day but,simultaneously, wants to provide an irrigation service to the2831 hydrants, then he should modify the maximum speedconstraint to obtain feasible solutions to the scheduling problem.Next, he can compute the smallest maximum speed required bythe system; in our particular case, there are points in the systemwhere the water flow speed is 3.68 m/s. Then, he can solve theirrigation scheduling problem with only three periods per day byfixing the maximum speed at a value above 3.68 m/s.
The running times for each optimization approach are shownin Table 2 for a real-life instance whose dimensions are given inTable 1. The headings are as follows: m, number of constraints; nc ,number of continuous variables; n01, number of 0–1 variables; nel,number of nonzero elements in the constraint matrix; and dens,matrix density. Notice the large dimensions of the mixed 0–1separable nonlinear model. See in Fig. 3 the pilot case’stopography.
The headings for Table 2 are as follows: Z, objective functionvalue; tt, total elapsed time; GAP, optimality gap with respect toZCPLEX , where it is the (optimal) solution value obtained by CPLEXv11.2 and GAP¼ ðZ�ZCPLEXÞ=ZCPLEX . Notice that Z for HEUR is only4.72% greater than the optimal solution value, what is totallyacceptable by the end-user, and the elapsed time (1.7 min) is veryaffordable. See in Section 5.3 the results of an extensivecomputational experience that have been carried out to validatethe HEUR algorithm for irrigation scheduling purposes and theLAZY algorithm for irrigation planning and simulation purposes.
ARTICLE IN PRESS
Fig. 3. Pilot case’s topography.
Table 2Computing effort for different optimization approaches.
Optimization model Machine Processor (GHz) Ram (Gb) Z tt GAP (%)
M_ETC (CPLEX based approach) SUN W2100 Opteron 2.6 4 2182.89 38 s –
M_ETC (LAZY heuristic) PC Pentium 1.6 2 2198.36 21 min 0.71
M_ETC (HEUR heuristic) PC Pentium 1.6 2 2285.99 1.7 min 4.72
MM_S (GLPK based heuristic) PC Pentium 1.6 2 2.23 m/s 36 s
MM (GLPK based heuristic) _P (GLPK based heuristic) PC Pentium 1.6 2 68.53 mca 2 s
Results of the pilot case study.
M. Alminana et al. / Omega 38 (2010) 492–500496
5.1. LAZY, heuristic optimization approach for irrigation planning
We have implemented the algorithm described in [1] forobtaining the optimal solution over a Windows XP platform withthe free software GLPK (see http://www.gnu.org/software/glpk/).Due to the complexity of the model M_ETC, the optimizationengine GLPK did not provide the results in an affordablecomputation time. Therefore, we developed the LAZY algorithm,a GLPK-based heuristic approach in order to solve the problem. Itcan be summarized as shown in the following steps (see thediagram in Fig. 4):
1.
Solve the approximate linear relaxation ðR_M_ETCÞ of thecontinuous nonlinear model of the irrigation schedulingproblem ðM_ETCÞ, where the linear Taylor series expansionapproximation of the quadratic variables is used and theintegrality of the 0–1 variables is also relaxed.2.
Update the water volumes for the solution that has beenobtained.3.
If the linear solution is a solution for the nonlinear continuousproblem, then we go to the next step. Otherwise, update theapproximation point of the Taylor series to the nonlinearmodel and go to step 1.4.
Fix to 1 the relaxed 0–1 y variables that have value 1 in thesolution of the linear relaxation.5.
Solve the integer linear approximation of the integer nonlinearproblem. Notice that the model only considers the y variableswith a fractional value in the previous step.6.
Update the water volumes for the solution that has beenobtained.7.
If the integer solution is a solution of the integer nonlinearproblem, then stop. Otherwise, update the approximationpoint to the nonlinear model and go to step 5.The optimality GAP is 0.71% for the pilot case (see Table 2). It isvery good but the elapsed time is 21 min. This time is valid forplanning and simulation studies but is not affordable for dailyscheduling work. The models MM_S and MM_P are optimized byusing the same GLPK-based approach.
5.2. HEUR, heuristic approach for irrigation scheduling
We have also implemented in the DSS WISCHE a fasterheuristic algorithm, so-called HEUR, for obtaining an acceptablesolution in an affordable computing time.
ARTICLE IN PRESS
1. Solve R − M − ET C
2. Update flow volumeCalculate the friction factorModify the Linear Problem
3. Feasibility ?
4. IF yht = 1 → yht := 1IF yht < 1 → yht ∈ {0, 1}
Modify the linear integer problem
5. Solve the linear integer problem
6. Update water volumeCalculate the friction factorModify the Integer Problem
7. Feasibility ?
Solution
yes
yes
no
no
Fig. 4. LAZY heuristic optimization approach for planning and simulation studies.
M. Alminana et al. / Omega 38 (2010) 492–500 497
The DSS basically works as follows in its daily basis: At theprevious day and a given hour, there is a set of hydrants that havebeen provisionally assigned for irrigation, and a set of hydrantsapply for irrigation assignment. Next, the system performs theassignment for the new arrivals and, if the satisfaction of theconstraints and/or the objective function improvement need it, italso modifies the provisional assignment of the previously alreadyscheduled hydrants. The execution is been carried out until thehour limit of the day is reached. At that time, the provisionalassignment is considered definitive.
The algorithm can be summarized in the following steps (seethe diagram in Fig. 5):
1.
Order the non-yet assigned hydrants by the non-decreasingirrigation earliness–tardiness unit cost divided by the re-quired water volume. Let A denote the set of those hydrants,and let A denote the set of hydrant that cannot been assignedyet.2.
For the hydrant, say, h with the smallest weighted assignmentcost for the irrigation time period t, call the functionevaluateyht . This function updates the water volume fromhydrant h to its head node, analyzing whether it is feasible toassign hydrant h to time period t, i.e., whether the pressureand speed constraints are satisfied along the path fromhydrant h to its head node.3.
If hydrant h can irrigate during time period t, then we updateset A, such that A :¼ A�fhg and go to step 10. Otherwise,update time period t, such that t : ¼ tþ1.4.
If there are still irrigation time periods to assign to hydrant h,then go to step 2.5.
Otherwise, remove the hydrants which are ancestors ofhydrant h and have higher weighted assignment unit cost,such that their accumulate water volume exceeds the watervolume required by hydrant h. Let us call Ph to this set ofhydrants. Notice that, in such a case, the assignment ofhydrant h to its lowest weighted assignment irrigation timeperiod is feasible.
6.
If hydrant h can be assigned to the irrigation time period t,then try to assign the previously removed hydrants includedin set Ph. Let P0h � Ph be the subset of hydrants that cannot beassigned again and, then, update A :¼ ASP0h and A : ¼ A�fhg,
and go to step 10.
7. Otherwise (i.e., the hydrant h cannot be assigned), restore theremoved hydrants set Ph and update set A.
8. If Aa|, it means that there are still untested hydrants, thenmove to the next hydrant, if any, in the order of the priorityvalue, i.e, update h : ¼ hþ1 and go to step 2.
9.
If nitero2 then restore the network, reset niterþþ , andassign the not yet assigned hydrants in the previous iteration.Update all the sets and start the algorithm at Step 2.Otherwise, stop since a solution has not been found.10.
If Aa| it means that there are unassigned hydrants, then gotostep 2. Otherwise, goto step 11.11.
If A¼ | it means that there are not anymore unassignedhydrants, then the solution has been found. Otherwise, go tostep 9.Since only two major iterations are allowed, it is easy to verifythat the complexity of HEUR is OðH2Þ, where H is the number ofhydrants in the network under consideration.
Table 2 shows the time (1.7 min) and the optimality GAP(4.72%) of HEUR for the pilot case, what is fully affordable forirrigation scheduling work. In [21] some heuristic algorithms fortotal tardiness minimization in a flowshop problem are presentedobtaining good results for a constructive heuristic based in jobpreferences.
5.3. Additional computational experience
In this section we show the results of the computationalexperiment for obtaining different solutions of the algorithms fora set of instances for the pilot case. We vary the number ofirrigation time periods and the assignment earliness–tardinessunit costs without modifying the case’s topology. The computa-tional experience was executed on an Intel Core Duo 1.66 GHzprocessor, with 2 Gb RAM running under Microsoft Windows XPoperating System.
Table 3 shows the solution for the M_ETC model usingWISCHE. The headings are as follows: nper, number of periodsin the daily scheduling; ZLAZY and ZHEUR, solution value byusing the LAZY and HEUR approaches, respectively; GAP¼
ðZi�ZCPLEXÞ=ZCPLEX , where i¼ L for LAZY and H for HEUR; vmax,solution of the MM_S problem, the maximum water speedallowed for the scheduling problem.
Notice that the network can only satisfy the water demand forall the hydrants when five or six time periods are considered. Forless than five periods we need to minimize the maximum toobtain a feasible water speed in the network. On the other hand,see that the LAZY solution value does not differ from the optimal(CPLEX) value in more than GAP¼ 0:80%, while the HEUR solutionvalue is not greater than 6.3% from the optimal value in the worstcase. On the other hand, the elapsed time is very affordable.
In order to confirm the validity of the WISCHE DSS, we carriedout an additional computational study by perturbing the topologyof the real-life case under consideration. Table 4 shows the mainresults, where the heading ‘‘network size’’ gives the number ofhydrants considered in the related cases. We can observe thesuperb results that have been obtained. Fig. 6 depicts the
ARTICLE IN PRESS
Fig. 5. HEUR, heuristic approach for irrigation scheduling.
Table 3Computational effort for scheduling purposes in the pilot case.
Case nper ZCPLEX tt (s) ZLAZY tt GAPL (%) ZHEUR tt (s) GAPH (%) vmax (m/s)
C1 2 1585.29 16 1586.39 57 s 0.07 1593.30 21 0.51 5.30
C2 2 Infeasible – Infeasible – – Infeasible – – 2.50
C3 3 1902.85 24 1918.03 2.5 min 0.80 1958.48 55 2.92 3.70
C4 3 Infeasible – Infeasible – – Infeasible – – 2.50
C5 4 2038.50 30 2053.91 6.3 min 0.76 2149.12 75 5.43 2.78
C6 4 2186.34 32 Infeasible – – Infeasible – – 2.50
C7 5 2247.08 37 2263.96 39.6 min 0.75 2388.87 140 6.30 2.23
C8 5 2182.89 38 2198.36 21.8 min 0.71 2285.99 104 4.72 2.50
C9 6 2351.10 42 2368.03 37.9 min 0.72 2475.96 150 5.31 1.84
C10 6 2182.33 43 2196.69 13.2 min 0.66 2205.43 13 1.06 2.50
M. Alminana et al. / Omega 38 (2010) 492–500498
ARTICLE IN PRESS
Table 4Computational effort for scheduling purposes in a realistic testbed of networks.
Network size nper ZCPLEX tt (1) (s) ZLAZY tt (2) (s) GAPL (%) ZHEUR tt (2) (s) GAPH (%)
100 2 78.34 2.23 78.48 0.77 0.18 78.86 0.52 0.66
100 3 91.51 4.32 92.79 4.28 1.40 92.91 0.55 1.53
100 4 91.15 5.63 92.24 4.70 1.20 92.88 0.49 1.90
100 5 93.56 5.91 95.32 8.92 1.88 95.17 0.55 1.72
100 6 94.92 7.94 96.66 19.81 1.83 96.24 0.61
250 2 133.02 2.29 134.08 13.75 0.80 133.96 0.95 0.71
250 3 147.26 4.23 150.41 5.46 2.14 149.20 1.01 1.32
250 4 155.07 8.49 157.37 26.32 1.48 160.95 1.39 3.79
250 5 159.41 24.71 161.89 74.84 1.56 162.23 0.95 1.77
250 6 166.37 20.96 168.70 190.16 1.40 168.91 1.16
500 2 250.37 11.80 251.43 2.53 0.42 251.89 2.64 0.61
500 3 288.73 10.77 293.20 24.02 1.55 293.18 4.28 1.54
500 4 302.53 9.16 307.04 22.17 1.49 306.36 1.61 1.27
500 5 348.15 308.23 353.33 4206.19 1.49 357.82 3.67 2.78
500 6 349.19 979.55 354.50 2145.16 1.52 355.83 2.23
1000 2 518.99 9.77 520.31 3.97 0.25 520.31 3.22 0.25
1000 3 610.67 115.50 614.85 34.24 0.68 618.96 7.13 1.36
1000 4 672.05 478.81 680.40 339.69 1.24 695.77 8.25 3.53
1000 5 689.34 288.31 696.51 8613.22 1.04 697.70 3.61 1.21
1000 6 698.17 93.69 705.79 4295.41 1.09 708.57 4.30
2000 2 1183.15 10.12 1183.65 3.61 0.04 1188.64 15.08 0.46
2000 3 1373.46 150.24 1386.89 68.72 0.98 1406.78 30.16 2.43
2000 4 1467.71 190.99 1477.13 107.22 0.64 1549.43 52.66 5.57
2000 5 1572.48 250.47 1582.36 1616.66 0.63 1657.50 99.05 5.41
2000 6 1622.81 290.90 1632.83 14400.00 0.62 1697.07 31.47
2831 2 1585.29 150.80 1586.39 57.49 0.07 1593.30 21.13 0.51
2831 3 1902.85 24.24 1918.03 151.20 0.80 1958.48 55.84 2.92
2831 4 2038.50 30.13 2053.91 377.74 0.76 2149.12 75.64 5.43
2831 5 2182.89 38.07 2198.36 1306.61 0.71 2285.99 104.61 4.72
2831 6 2351.10 42.31 2368.03 2273.36 0.72 2475.96 150.89 5.31
(1) Sun W2100, Opteron 2.6 GHz processor, 4 Gb of Ram. (2) PC Pentium Intel Core Duo 1.66 GHz processor, 2 Gb of Ram, Windows XP.
0
1
2
3
4
5
LAZY HEUR
GA
P%
Fig. 6. GAP % for results shown in Table 4.
M. Alminana et al. / Omega 38 (2010) 492–500 499
minimum, average and maximum GAP obtained by the LAZY andHEUR algorithms in the whole testbed.
6. Conclusions
The DSS WISCHE is an application built jointly with Riegos deLevante (the main group from the Agriculture Community ofElche, ACE), irrigation community which has a real need for
irrigation scheduling because water demand usually exceed wateravailability. The DSS allows irrigation community managers toplanning hydrant turns one week in advance and to schedulinghydrant turns one day in advance. Three models are considered inthe DSS, namely, M_ETC for minimizing the irrigation earliness-tardiness cost, MM_S for minimizing the maximum water speedand MM_P for minimizing the maximum water pressure.Although the mixture of models is considered for irrigationsimulation, planning and scheduling study purposes, M_ETC is themodel of choice for daily scheduling work. Two heuristicalgorithms have been presented for obtaining acceptable solu-tions for the large-scale mixed 0–1 separable nonlinear modelMPF_ETC, namely, LAZY and HEUR. Although CPLEX achieves theoptimal solution and takes less computational time, this approachwas not consider because of its economic cost and the reasonableresults provided by the algorithms LAZY and HEUR as it has beenshown. Given the acceptable solution quality and the goodcomputing time in a set of experiments with a pilot case, theuser has decided to utilize LAZY for planning purposes and HEURfor scheduling work. Scheduling is as flexible as possible and thesystem presented allow the farmers to be serviced according totheir previous usage, so farmers who make early applications andusually make reasonable use of their demand benefit from theproposed decision support system.
Acknowledgments
The authors like to thank the three anonymous referees fortheir helpful criticism and suggestions, they have significantly
ARTICLE IN PRESS
M. Alminana et al. / Omega 38 (2010) 492–500500
contributed to the improvement of an earlier version of the paperand the Decision Support System implementation.
References
[1] Alminana M, Escudero LF, Landete M, Monge JF, Rabasa A, Sanchez-Soriano J.On a mixed 0–1 separable nonlinear approach for water irrigation scheduling.IIE Transactions 2008;44(4):398–405.
[2] Andre FJ, Herrero I, Riesgo L. A modified DEA model to estimate theimportance of objectives with an application to agricultural economics.Omega 2009;38(6).
[3] Andreu J, Capilla J, Sanchis E. AQUATOOL, a generalized decision-supportsystem for water-resources planning and operational management. Journal ofHydrology 1996;177:269–91.
[4] Anwar AA, Clarke D. Irrigation scheduling using mixed-integer linear program-ming. Journal of Irrigation and Drainage Engineering 2001;127(2):63–9.
[5] Baker KR, Scudder GD. Sequencing with earliness and tardiness penalties: areview. Operations Research 1990;38(1):22–36.
[6] Balabanis P, Peter D, Ghazi A, Tsogas M, editors. Mediterranean desertifica-tion: research results and policy implementation, two volumes. EuropeanCommission, 2000.
[7] Brucker P. Scheduling algorithms. Berlin: Springer; 2001.[9] Day JM, Wright PD, Schoenherr T. Improving routing and scheduling
decisions at a distributor of industrial gasses. Omega 2009227–37.[10] DeVries TT, Anwar AA. Irrigation scheduling. I. Integer programming
approach. Journal of Irrigation and Drainage Engineering 2004;130(1)9–16.
[11] DeVries TT, Anwar AA. Irrigation scheduling. II. Heuristic approaches. Journalof Irrigation and Drainage Engineering 2004;130(1):17–25.
[12] DeVries TT, Anwar AA. Irrigation scheduling with travel times. Journal ofIrrigation and Drainage Engineering 2006;132(2):220–7.
[13] Escudero LF. WARSYP, a robust approach for water resources planning underuncertainty. Annals of Operations Research 2000;95:313–39.
[14] Labadie JW, Brazil IE, Corbu I, Johnson LF, editors. Computerized decisionsupport systems for water managers. American Society of Civil Engineers,1989.
[15] Loucks DP. Water resource systems models: their role in planning. Journal ofWater Resource Planning Management 1992;118.
[16] Loucks DP, da Costa JR, editors. Decision support systems. Water resourcesplanning. Berlin: Springer; 1991.
[17] Marco JB, Harboe R, Salas JD, editors. Stochastic hydrology and its use inwater resources systems simulation and optimization. Dordrecht: KluwerAcademic Publishers; 1993.
[18] Pereira MVF, Pinto LMVG. Multistage stochastic optimization applied toenergy planning. Mathematical Programming 1991;52:359–75.
[19] Pinedo M. Scheduling. Theory, algorithms and systems. Englewood Cliffs, NJ:Prentice-Hall; 1995.
[20] Recio B, Ibanez J, Rubio F, Criado JA. A decision support system for analysingthe impact of water restriction policies. Decision Support Systems2005;39:385–402.
[21] Ronconi DP, Henriques LRS. Some heuristic algorithms for total tardinessminimization in a flowshop with blocking. Omega 2009272–81.
[22] Wolsey L. Integer Programming. New York: Wiley; 1998.