International Journal of Advancements in Research & Technology, Volume 3, Issue 10, October ISSN

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1 Itertol Jourl of Avceets Reserch & echology, Volue 3, Issue, October -4 5 ISS Mofe Super Coverget Le Seres Metho for Selecto of Optl Crop Cobto for Itercroppg Iorey Etukuo Deprtet of Sttstcs, Uversty of Clbr, Clbr, ger. El: seobogorey@gl.co ABSRAC Frers re geerlly cofrote wth probles of eterg optl crop cobtos whe terctve effects re preset betwee crops grow tercroppg schee. hs, therefore fors the vryg rg of proftblty frg eercse. I orer to frers obt optu yel, ths pper proposes copletely ew etho to optlly select crop cobtos for ther tercroppg schee. uercl llustrto gve shows tht the etho s equte for ths purpose. Keywors : Optl esgs, ofe super coverget le seres, qurtc progrg, crop cobtos, frl. IRODUCIO rsg fro the hgh populto growth rte the ee for crese foo proucto, both sll lrge scle A A QUADRAIC PROGRAMMIG MODEL FOR frers re fce wth the proble of selecto of SELECIG OPIMAL CROP COMBIAIOS I optl crop cobtos for ther tercroppg schee tht IERCROPPIG wll yel u proft. Owuee Sh [] he qurtc progrg oel for crop cobtos Igbozurke [] efe tercroppg schee s elberte tercroppg schee s gve s follows prctce of cultvtg two or ore crops sulteously o the se prcel of l. Mze f ( ) he prctce of tercroppg s ore populr the c + q k k k ecooclly uer evelope tos occupes bout ety percet of croppe re ost coutres, prtculrly subect to the ropcl R Forest Se Ar ropcs, [3] [4]. Accorg to [], there s yel vtge growg crops together rther th growg ech oe seprtely becuse of l the fct tht crops copleet oe other ther use of fel L te. Ag, the spre of sese pests s coserbly less rp tercroppg th sole croppg. s Etukuo Uore [5] hve lrey evelope S qurtc progrg oel to solve ths proble the soluto techque opte ws the ofe sple etho. f F owever, ew lgorth kow s ofe super coverget le seres (MSCLS Q) hs bee evelope for solvg qurtc progrg probles, [6]. Mewhle, p P Etukuo Uore [7] copre the two ethos of qurtc progrg probles ely, ofe sple etho (MSM) ofe super coverget le seres M etho (MSCLS Q) coclue tht MSCLSQ etho s ore effcet th MSM hlg qurtc progrg v V probles bse o well kow esures of effcecy of lgorth. hs pper, therefore focuses o the MSCLSQ h pproch eterg the optl selecto of crop cobtos tercroppg schee. he super coverget le seres lgorth s le serch lgorth whch kes g G use of the prcples of optl esgs of eperet to get to the optzer. B,,,,,,,, Copyrght 4 ScResPub. ()

2 Itertol Jourl of Avceets Reserch & echology, Volue 3, Issue, October -4 5 ISS where l cost of preprto of l per hectre wth respect to crop,,,..., s cost of sees/seelgs per hectre of crop,,,..., f cost of fertlzer eee per hectre of crop,,,..., p cost of pltg per hectre wth respect to crop,,,..., cost of fr geet fro pltg to hrvestg wth respect to crop,,,..., v cost of hrvestg per hectre wth respect to crop,,,..., g cost of surce cover per hectre of crop,,,..., h cost of post hrvest hlg per hectre of crop,,,..., L totl fus vlble for preprg frl for tercroppg schee S totl fus vlble for purchse of seelgs wth respect to ll crops the tercroppg schee F totl fus vlble for procureet of fertlzer for ll the crops the tercroppg schee P totl fus vlble for pltg ll the crops the tercroppg schee M totl fus vlble for geet of the tercroppg schee fro pltg to hrvest te V totl fus vlble for hrvestg of ll the crops the tercroppg schee G totl fus vlble for obtg surce cover for ll the crops the tercroppg schee totl fus vlble for post hrvest hlg of the tercroppg schee Step : Gve the respose surfce f() c + ½ Q () Select support pots such tht k( + ) k( + ) + k, where k s the uber of prttoe groups esre s the uber of vrbles. ece, by rbtrrly choosg the support pots s log s they o ot Step : Step 3: volte y of the costrts, ke up tl esg tr (3) Prtto to k groups wth equl uber of support pots obt the esg tr,,,,, k for ech group. Obt the forto trces M,,,, k ther verses M,,,, k Copute the trces of the tercto effect of the vrbles for the groups, I where,,..., k the vector of the tercto preters obte fro f() s gve by q g q. q3 he tercto vectors for the groups re gve by I M g the trces I of the e squre error of the groups re 3 MEODOLOGY M M + I I. Step 4: he trces of coeffcet of cove he sequetl proceure for the MSCLSQ gve below requres tht the optl support pots tht for the tl cobtos of the trces of e squre error re esg tr obte fro the etre eperetl rego be prttoe to k groups, k, 3, so tht optl v v v 33 g strtg pots re obte for ech group. owever, [8],, v v v showe tht wth k for qurtc progrg probles, 33 optl solutos re obte. he sequetl steps volve MSCLSQ re gve s follows: g{h, h, h3},,, k (4) By orlzg such tht I, we hve g h h, h h, h 3 h 3 (5) he verge forto tr s gve by M(ξ) Copyrght 4 ScResPub.

3 Itertol Jourl of Avceets Reserch & echology, Volue 3, Issue, October ISS Step 5: Step 6: (6) Fro f(), obt the respose vector where z z z z, z z f(,,..., ) 3 [ + ] z f(,,..., ) 3 [ + ]... z f(,, [ + ] 3[ + ]..., ) [ + ][ + ] ece, we efe the recto vector Step 8: Mke ove to the pot M - ( ξ )z (7) ρ () Step 9: Copute f( ) f( ). Is f( ) - f( ) < ε where ε., the stop for the curret soluto s optl, otherwse, replce by retur to step 7. If the ew step legth, ρ s eglgbly sll, the the optzer h bee locte t the frst ove (8 4 PROCEDURE FOR OBAIIG A OPIMIZER USIGE MEOD he ssupto here s tht the sol lyss of the frl to be use for the tercroppg schee h bee crre out wth respect to ll the crops to be clue the schee tht the crops c thrve very well bse o the lyss. by orlzg such tht, we hve ) Copute the optl strtg pot, fro u ; u > ; u u,,,, Step 7:,,,,. Obt the step legth, ρ fro A b A ρ (9) for zto proble or fro A b A ρ () for zto proble, A b where,,,, s the th costrt of the qurtc progrg proble. If there re t crops to be cultvte o the frl, there t re t C possble groups of crops fro whch the optl cobto c be selecte where s the uber of crops to be tke fro t for cobto. As llustrto, let us ssue tht we hve four crops, ely ze, y, pepper okro eote respectvely s crops,, 3 4 for ths eercse. We ssue further tht sol lyss fvours the four crops o prcel of l cqure for frg. he t for ths llustrto obte fro [5] re gve bles 5. Copyrght 4 ScResPub.

4 Itertol Jourl of Avceets Reserch & echology, Volue 3, Issue, October ISS ble : Cost of ech frg operto per crop per hectre Cost ( ) Crops () 3 4 l s f p v 4 3 h 5 3 g 3 4 ble : he per hectre effects of oe crop the other whe plte together Crop () Subect to , We ow obt the soluto of QP by super coverget le seres etho s follows: Step Let be the re efe by the costrts. ece, {, ; C} Select support pots such tht k (+) k (+) + k, ble 3: Optl coeffcet syetrc tr of per where k s the uber of prttoe groups esre s the hectre effects of oe crop the other uber of vrbles. By choosg k, we hve Crop ece, by rbtrrly choosg 6 support pots s log s they o ot volte the costrts, the tl esg tr s ble 4: Vlue of resource (oetry) costrts (3) Resource Vlue ( ).3.4 L..5 S 5 F..6 P 6 M 8 Step : V 3 Prtto to two groups such tht 5 G.6. ble 5: Epecte proft per crop per hectre.5. Crop Proft ( ) A 6 hectres he ecso vrbles re hectres of l llocte to crop,,,...,.3.4 k hectres of l llocte to crop k, k,,..., Fro the four vlble crops, there re eleve fferet..5. cobtos such s (, ), (, 3), (, 4), (, 3), (, 4), (3, 4), (,,..6 3), (,, 4), (, 3, 4), (, 3, 4), (,, 3, 4). Substtutg the t for ech of the cobtos, qurtc progrg oel (QP) for he forto trces re M M cobto (, ) s gve by ther verses re respectvely - - M f() [ ] M Copyrght 4 ScResPub.

5 Itertol Jourl of Avceets Reserch & echology, Volue 3, Issue, October ISS M Step 3: Obt the trces of coeffcets of cove cobtos fro M M. hese re g{.857,.857,.857} Slrly, (4) ,.8,.8, 3 4 I g{.857,.857,.857} (5) - - by orlzg such tht.775,.799 I,,, 5 6 we hve 4.6 g{.69,.69,.69} (6).7576 Step 3: Obt the optl strtg pot, s follows: [.6.].6.37,. g{.976,.976,.976} (7) he verge forto tr s gve by M(ξ ) by orlzg such tht, we hve Copyrght 4 ScResPub Step 4: Fro f(), obt the respose vector z z z z ece, we efe the recto vector M - (ξ )z Sce u,,,,, the u.583, u.68, u.735, 3 u.68, u ece, the optl strtg pot s u Step 6: Obt the step legth, ρ fro u.735, 4

6 Itertol Jourl of Avceets Reserch & echology, Volue 3, Issue, October ISS A b A ρ (8) where A b,,,, s the th costrt of the ler progrg proble. For ρ A [ 9] 8 b, we hve.3499 [ 8 9] [ 8 9] ble 6: Sury of optl soluto obte by MSCLS Q etho Slrly, the step legths for the reg costrts re Moel Crop Vlue of ecso Obectve fucto respectvely.53,.946,.54, , Cobto vrbles vlue 7.67,.36, 3.989,.383,.33. QP, 565.4, 75,94. 6 We choose the u step legth, ρ.33. QP, , 59, QP 3, 4 6, 56,665.6 Step 7: Mke ove to the pot QP 4, 3 6, 75, QP 5, , 6, QP 6 3, 4 3 6, 54,87.56 ρ QP 7,, 3 6, 99, , 3 (.33) QP 8,, , 74, , QP 9, 3, , 7,977.8 Step 8: ow, f( ) 6(.5654) + 7(.6) f( ) 6(.3499) + 7(.3499) Sce f( ) - f( ) , ke seco ove by replcg by By usg the costrt tr tht gve the u ρ, we obt ρ s follows:. Sce ρ.5654 [ ] [ ] ρ, the the optzer ws locte t the frst ove, hece, f( ) By kg slr coputtos, we hve the results s splye o ble 6 below for ll the crop cobtos , QP, 3, 4-56., , QP,, 3, , 79.7, 3-49., 4 6 Moels QP QP re ot perssble sce the vrbles ust tke oly postve vlues ot greter th 6. ece, the optl soluto s obte fro oel QP7 wth the obectve fucto vlue of 99, COCLUSIO he prry obectve of ths stuy, ely, optl selecto of Copyrght 4 ScResPub.

7 Itertol Jourl of Avceets Reserch & echology, Volue 3, Issue, October ISS crop cobtos tercroppg schee by ofe super coverget le seres etho hs bee successfully crre out. As coul be see ble 6, tercroppg schee cosstg of crops, 3 yels the hghest proft of 99,589.8 followe by the schee cosstg of crops wth proft of 75,94.. herefore, orer to hve u proft for hs frg busess, the frer shoul opt tercroppg schee cosstg of crops, 3 cultvte 6 hectres of ze, 59.3 hectres of y 38.9 hectres of pepper o the se frl. REFERECES [] I.C. Owuee.D. Sh (99): Fel crop Proucto opcl Afrc. echcl Cetre for Agrculturl Rurl Cooperto, Ee, he etherls5 pp [] U.M. Igbozurke (977): Agrculture t cross ros, Uversty of Ife Press, ger, pp [3] R. Mee J. Rley (98): A revew of sttstcl es relevt to tercroppg reserch, Jourl of Sttstcl Socety, 44(4), pp [4] S. Wk,. Fwus D. u (98): Foo gr yels fro tercroppg ze Vg Ugucult (L) Wlp, ger Jourl of Agrculturl Scece, 99, pp [5] I.A. Etukuo M.U. Uore (3): A qurtc progrg oel for crop cobtos tercroppg, Globl Jourl of Mthetcl Sceces, Vol, o., pp [6] M.U. Uore I.A. Etukuo (9): A ofe super coverget le seres lgorth for solvg qurtc progrg probles, Jourl of Mthetcl Sceces, Vol., o., pp [7] I.A. Etukuo M.U. Uore (9): A coprso of ofe super coverget le seres lgorth ofe sple etho for solvg qurtc progrg probles, ICASOR Jourl of Mthetcl Sceces, Vol. 3, o., pp [8] P.E. Chgbu.A. Ugbe (): O the segetto of the respose surfces for super coverget le seres optl solutos of costre ler qurtc progrg probles, Globl Jourl of Mthetcl Sceces, Vol., o. &, Copyrght 4 ScResPub.