Estimation Using Differential Evolution for Optimal Crop Plan

Transcription

1 Estmaton Usng Dfferental Evoluton for Optmal Crop Plan Mlle Pant, Radha, Deept Ran 2, Ath Abraham 3 and D. K. Srvastava 4 Department of Paper echnology, II Roorkee, Inda. 2 Unversty of Évora, úcleo da Mtra, , Évora 7000 Évora, Portugal 3 Q2S, orwegan Unversty of Scence and echnology, orway. 4 Department of Hydrology, II Roorkee, Inda. mllfpt@tr.ernet.n, t.radha@eee.org, deeptnatyan@yahoo.com, ath.abraham@eee.org, dkshyfhy@tr.ernet.n. Abstract. hs paper presents an applcaton of Dfferental Evoluton (DE) to determne optmal crop plan for command area of Pamba-Achankovl-Vappar (PAV) lnk proect, so as to maxmze the net rrgaton beneft. he mathematcal model of the problem s lnear n nature subect to varous constrants due to avalablty of total land area, water, fertlzers, seeds and manure, etc. umercal results show that DE gves a better performance n comparson to the usual software tools used for solvng such problems. Keywords: Dfferental Evoluton, Reservor, Crop Plan, Optmzaton. Introducton DE has emerged as one of the most promsng contemporary optmzaton technque n the past few years. Some of the reasons for the popularty of DE nclude easy mplementaton, lttle parameter tunng and fast convergence. It has been successfully appled to solve a wde range of optmzaton problems such as clusterng [], unsupervsed mage classfcaton [2], dgtal flter desgn [3], optmzaton of nonlnear functons [4], chemcal engneerng processes [5] and mult-obectve optmzaton [6] etc. In the present study we dscuss the performance of DE to obtan optmal crop plans under adequate, normal and lmted rrgaton water avalablty for rrgaton area under the PAV lnk proect. A lnear programmng based optmzaton model s used for ths. he proposed Pamba-Achankovl-Vappar Lnk proect has three storage reservors, two tunnels, necessary canal system and a few power generatng unts [7]. he Punnamedu reservor (reservor-2) s located at a hgher elevaton on rver Pamba Kal Ar n the Pamba basn of Karala state, whch serves a part/full of ts downstream mandatory requrements and supples surplus water to reservor- by ntra-basn export of surplus water (dverson) through tunnel-2. he Achankovl Kal Ar reservor (reservor-) located on Achankovl Kal Ar rver n Achankovl rver basn of Kerala state, supples water for rrgaton purposes to the state of amlnadu, through tunnel- to the man canal. he water from man canal s then dstrbuted to the command area of Vappar basn n amlnadu state. he reservor s proposed as a wthn-the-year storage scheme. Fgure shows a schematc representaton of the PAV lnk dverson system. Besdes ths, reservor- releases water for power

2 generaton. he Achankovl reservor (reservor-3), whch s located on Achankovl rver n the Achankovl rver basn of Kerala state, besdes actng as a pumped storage scheme accommodatng the water drawn from the upstream reservor-, also serves the purpose of releasng water to downstream to meet ts downstream mandatory demands. Also, f there s defct at reservor- the surplus water of reservor-3 can be pumped back to reservor-. he monthly nflow at reservor- for the 50 percent, 75 percent and 90 percent water year dependable flows are shown n Fgure 2. he GCA (gross command area) potental and CCA (culturable command area) potental of the proect would be 45,573 and 0,555 ha, respectvely. he proposal s to rrgate 9,400 ha (CCA actually consdered) of area per annum wth an rrgaton ntensty of 90 percent. he proposed croppng pattern was formulated by the amlnadu State Agrculture Department exclusvely for ths proect. Crop areas gven n the report were determned on the bass of the food requrements of the populaton lkely to be benefted from proect. he suggested croppng pattern conssts of 8 crops namely; Paddy, Olseed, Jowar, Vegetables (Brnal, Ladyfnger and Beans), Pulses, Bara, Cotton and Chlles; the proposed correspondng area allocaton for each crop s 5234, 709, 287, 5233, 6093, 5233, 287 and 824 ha, respectvely, and crop yelds from these crops under rrgaton area; 5.39,.5, 2.56, 3.0, 0.74, 2.56,.66 and.5 metrc tones (M.) per unt cropped area, respectvely. he gross rrgaton requrement for the command area as per proect s 635 mcm ncludng transmsson losses. he total producton from proposed rrgaton, cost of produce and expenses on cultvaton of varous crops under rrgaton condtons,.e., total cost on seeds, fertlzers, manure, rrgaton charges and labour per unt area of each crop s avalable from proect report [8]. he water release from reservor- for rrgaton s obtaned through ont operaton of the reservor system wth the help of successve approxmaton dynamc programmng model [9]. he water avalable at the man canal s further dstrbuted among the areas under Reaches-I, II and III. In ths study, however, optmal croppng patterns are presented for the total CCA for water avalable durng 50 percent, 75 percent and 90 percent water year dependable flows. he remanng of the paper s organzed as follows; n secton 2, we brefly descrbe the workng of DE. In secton 3, we dscuss the mathematcal model used n the present study. Secton 4 deals wth expermental settngs. In secton 5, we gve numercal results and the paper fnally concludes wth secton 6. Fgure 2 Monthly nflows at Reservor-I

3 Pamba Kal Ar rver Achankovl Kal Ar rver 2 Inflow Punnamedu reservor unnel - 2 Inflow Achankovl Kal Ar reservor downstream release or spll PH - 2 downstream release or spll unnel - Man canal Reach-I Reach-II Reach-III o sea PH - Study Area o sea al water PH Achankovl reservor Inflow Achankovl rver Outlet Reservor Powerhouse unnel Fgure Schematc Dagram for PAV Lnk Dverson System 2 Dfferental Evoluton Dfferental Evoluton was proposed by Storn and Prce [0]. It s a populaton based algorthm lke genetc algorthms usng the smlar operators; crossover, mutaton and selecton. he man dfference n constructng better solutons s that genetc algorthms rely on crossover whle DE reles on mutaton operator []. DE works as follows: Frst, all ndvduals are ntalzed wth unformly dstrbuted random numbers and evaluated usng the ftness functon provded. hen the followng wll be executed untl maxmum number of generaton has been reached. For a D-dmensonal search space, each target vector, a mutant vector s generated by v, g x, g + = xr, g + F *( xr, g xr, g ) () 2 3 where r, r2, r3 {,2,..., P} are randomly chosen ntegers, must be dfferent from each other and also dfferent from the runnng ndex. F (>0) s a scalng factor whch controls the amplfcaton of the dfferental evoluton x x ). In order to ( r, g r, g 2 3 ncrease the dversty of the perturbed parameter vectors, crossover s ntroduced [2]. he parent vector s mxed wth the mutated vector to produce a tral vector u, g+,

4 v, g + f ( rand u, g + = x, g f ( rand CR) > CR) or and ( ( where =, 2,, D; rand [0,] ; CR s the crossover constant takes values n the range [0, ] and rand (,2,..., D) s the randomly chosen ndex. Selecton s the step to choose the vector between the target vector and the tral vector wth the am of creatng an ndvdual for the next generaton. = rand rand ) ) (2) 3 Mathematcal Model A lnear programmng based optmzaton model s used for crop plannng. he model maxmzes net returns from crops and yelds optmal crop plan and monthly releases requred from reservor-. Surface water, land avalablty, fertlzers (, P, and K), seeds and manure requrements are consdered as constrants n the model. For the purpose of modelng, the crops have been segregated as food grans, cash crops and others. Paddy, Jowar and Bara are clubbed together as these falls under the category of food grans. Crop Plannng Model Obectve functon: Max Z = G P (3) = y b A and P = = = G (CS + CM + CF + CI + CLH ) A In the above equaton Z = annual net return from rrgated agrculture, G = total annual gross returns from crops, P = total annual net expenses on cultvatng crops, = total number of crops, =, 2,, 8 for Paddy, Olseeds, Jowar, Vegetables, Pulses, Bara, Cotton and Chlles, respectvely, A = area under th crop, CS = expenses on seeds for th crop per unt area, CM = expenses on manure for th crop per unt area, CF = expenses on fertlzers for th crop per unt area, CLH = expenses on labor and machnery for th crop per unt area, CI = expenses on rrgaton water charges for th crop per unt area, y = crop yeld n weght unts from th crop per unt area, and b = value of crop produce from th crop per unt yeld. Constrants () Surface Water Avalablty Constrant he volumes of water releases should be less than or equal to the surface water avalable,.e., = (W, t ) A R t for all t (4)

5 where W,t = gross rrgaton requrement of th crop durng tme perod t n terms of depth, A = area under th crop and R t = rrgaton water released from reservor n tme perod t. () Land Avalablty Constrants: he net area allocated to the gven crops should be less than or equal to the total area avalable,.e., = (λ,t )A A, for all t and A A (5) where λ,t = land use coeffcent of the th crop durng tme perod t and A = total area under rrgaton per annum. () Yeld Requrement Constrant: Yeld produced from the crops should be greater than or equal to the proposed yeld requrement,.e., = y A y where y for=,3 and6 = total yeld requred from and y A th crop. y forall (6) (v) Fertlzers Avalablty Constrants: Quantty of fertlzer of type ƒ requred wll be less than or equal to the quantty of fertlzer type ƒ avalable,.e., = Ff, A Ff, forall f (7) where F f, = quantty of fertlzer type f requred per unt area for th crop and F f, = total avalable quantty of fertlzer of type f. hree types of fertlzers have been consdered n the applcaton of model,.e., trogen, Phosphorus and Potassum (, P, K). (v) Manure Avalablty Constrant: otal quantty of manure requred wll be less than or equal to the total quantty of manure avalable,.e., = M A M (8) where M = quantty of manure requred per unt area for th crop and M = total avalable quantty of manure. (v) Seeds Avalablty Constrant: S A S for all (9)

6 where S = quantty of seeds requred per unt area for th crop and S, = total avalable quantty of seeds for th crop. (v) Bounds on Areas under Varous Crops: A,mn A A,max (0) where A,mn = lower lmt on the area under th crop and A,max = upper lmt on the area under th crop. A mnmum crop area constrant has been specfed on each crop so as to see that area occuped by crops does not fall below area under ran-fed cultvaton. It has also been specfed that area proposed under cotton and chlles should not be more than 8 percent and 7 percent of annual rrgaton. hs condton s ustfed because ther yelds have hgh revenues and optmally hgher area allocaton to these crops may cause reducton n food gran output, whch s socally undesrable. It has been consdered essental that total food gran producton should not be less than 0, 995 M.. 4 Expermental Settngs In ths secton we gve the data used for the mathematcal model used n secton 3 and the parameter settngs for DE. From nformaton avalable [3] estmates of average values of quanttes/ha requred for each crop for resource nputs,.e., seeds, manure and fertlzers are obtaned. otal requrements of these resources are obtaned from these values and crop area allocaton as per proect report. Intally t was assumed that total quantty avalable for each resource s equal to total quantty requred for the resource. he crop plannng model s solved usng LIGO package. he frst tral run s made of the model assumng that the amount of each resource avalable s equal to the requred amount, and from results t was seen that out of the total CCA,.e., 9400 ha only ha s allocated to the crops,.e., wth ths tral the total CCA was not allocated to varous crops (able ). Further model runs were made by varyng quantty of resource avalablty n some percent of requred amount the area allocatons for these trals are gven nable. Fnally t was assumed that 20 percent of the total quantty ntally estmated for each resource may be consdered as the extent of quantty avalable as nput, for whch almost all the area proposed has been allocated (able 2). Parameter settngs for DE: As dscussed n the earler secton, DE requres very few parameter settngs n comparson to other Evolutonary algorthms. Only three parameters; Populaton sze, crossover constant and scalng parameter are needed for DE. Populaton sze: 30; crossover constant CR= 0.5; Scalng parameter F = 0.5. Maxmum number of generaton: 000; umber of runs: 30;

7 able. Optmal area allocatons wth varable resource nputs avalable Resource Optmal Area Inputs Allocatons (ha) 80% % % % % % able 2. Extents of resource avalable Resource Fertlzers (kg) Extent Avalablty P K Manure (M..) 4335 Seeds (kg) Paddy Ol seeds Jowar Vegetables Pulses Bara Cotton Chlles umercal Results he performance of DE s compared wth LIGO, whch s another popular software tool for solvng Lnear Programmng Problems. From the numercal results gven n able 3, t can be seen that for 50% water year dependable flow, both DE and LIGO gave exact results. However for 75% and 90% water year dependable flow performance of DE s superor to LIGO n terms of net proft. For 75% water year dependable flow, the net proft as obtaned by DE s and by LIGO s , whch shows an mprovement of 0.33 %. For 90 % water year dependable flow DE gave a proft of Rs 539., whereas proft obtaned by LIGO s Rs , whch shows an mprovement of 0.36 %. Also we get to know the optmal area allocaton for dfferent crops whle usng DE and LIGO. he tme taken by DE and LIGO are same for all the three cases. 6 Concluson In the present study an applcaton of DE s shown for determnng the optmal croppng pattern so as to maxmze the net proft under the gven condtons. Its performance s compared vs-à-vs LIGO, whch s a very popular tool for solvng Lnear Programmng Problems. umercal results show that DE gves a superor performance n comparson to LIGO for 75% and 90% water year dependable flow. hus DE may be consdered as an alternatve for solvng such type of problems. In future we shall be comparng the performance of DE wth other technques lke PSO for solvng smlar types of problems.

8 . able 3. Performance Comparson of DE wth LIGO for 50%, 75% and 90% water year dependable flow Fg. (a) Fg. (b) Fg. (c) Fgures (a), (b), (c) represent convergence graphs for 50%, 75% and 90% water year dependable flow

9 Fg.2 Obectve functon value obtaned by DE and LIGO References. S.Paterln,.Krnk, Hgh performance clusterng wth dfferental evoluton. In: Proceedngs of the IEEE Congress on Evolutonary Computaton, vol. 2, 2004.pp M.Omran, A.Engelbrecht, A.Salman, Dfferental evoluton methods for unsupervsed mage classfcaton. In: Proceedngs of the IEEE Congress on Evolutonary Computaton, vol. 2, 2005a. pp R.Storn, Dfferental evoluton desgn for an IIR-flter wth requrements for magntude and group delay. echncal Report R , Internatonal Computer Scence Insttute, Berkeley, CA B.Babu, R.Angra, Optmzaton of non-lnear functons usng evolutonary computaton. In: Proceedngs of the 2 th ISME Internatonal Conference on Mechancal Engneerng, Inda, 200 pp R.Angra, B.Babu, Evolutonary computaton for global optmzaton of non-lnear chemcal engneerng processes. In: Proceedngs of Internatonal Symposum on Process Systems Engneerng and Control, Mumba, 2003.pp H. Abbass, A memetc pareto evolutonary approach to artfcal neural networks Lecture otes n Artfcal Intellgence, vol Sprnger, 2002a pp Ramakrshnan, M., Systems Analyss of Multreservors, M.E. Dssertaton, Dept. of Hydrology, Unv. of Roorkee, Roorkee, Inda, Feasblty Report of Pamba-Achankovl-Vappar Lnk Proect, WDA, Mnstry of Water Resources, ew Delh, Inda, Deept Ran, Multlevel Optmzaton of a Water Resources System, Ph.D. hess, Dept. of Mathematcs, Indan Inst. of ech., Roorkee, Inda, R. Storn and K. Prce, Dfferental Evoluton a smple and effcent adaptve scheme for global optmzaton over contnuous spaces, echncal Report, Internatonal Computer Scence Insttute, Berkley, D. Karaboga and S. Okdem, A smple and Global Optmzaton Algorthm for Engneerng Problems: Dfferental Evoluton Algorthm, urk J. Elec. Engn. 2(), 2004, pp R. Storn and K. Prce, Dfferental Evoluton a smple and effcent Heurstc for global optmzaton over contnuous spaces, Journal Global Optmzaton., 997, pp Handbook on Agrculture, ICAR, Mnstry of Agrculture, ew Delh, Inda, 997.