ࡱ> q`wbjbjqPqP 4::$+NNN8OO1DPS.S.S.SUUUDFFFFFF$5htjUUUUUj.S.ShVfVfVfUX.S.SDVfUDVfVfj$.S8P P/&N[X201cd$$tUUVfUUUUUjje^UUU1UUUU$&. . Jedna metoda procjene prinosa Dr. sc. Dominika Crnjac Mili University of Osijek, Faculty of Electrical Engineering, Kneza Trpimira 2b, HR  31000 Osijek, Croatia e-mail:  HYPERLINK "mailto:dominika.crnjac@etfos.hr" dominika.crnjac@etfos.hr Prof.dr.sc. Miljenko Crnjac University of Osijek, Faculty of Economics, Trg Lj. Gaja 7, HR-31000 Osijek, Croatia e-mail: crnjac@oliver.unios.hr Sa~etak Koristei saznanja matemati ke ekonomije i ekonometrije u radu emo pokazati alternative biranja modela. Dat emo neke efikasne metode za procjenu parametara Klju ne rije i: matemati ki model, prirast, funkcional, minimizacija, parametar Uvod U ratarskoj, sto arskoj i bilo kojoj drugoj proizvodnji vrlo je va~no dovoljno rano imati dobru procjenu prinosa. Ovaj problem promatrao je Singh 1992. godine. Pri tome je koriatena obi na logisti ka funkcija. Na osnovi podataka dobivenih od Agronomy Division of Indian Agricultural Research Institute, New Delhi, a koji su predstavljali koli inu suhe tvari i te~inu klasa po kvadratnom metru na poljima paenice i ri~e po evai od aestog tjedna nakon zasijana, pa sve do ~etve, procjenjena je gornja asimptota logisti ke funkcije. Na osnovi toga direktno se procjenjuje prinos. Ta procjena kasnije se mogla usporediti sa stvarnim prinosom nakon ~etve. U ovom radu ponudit e se viae alternativa biranja modela rasta i efikasna metoda za procjenu parametra koji predstavlja procjenu prinosa. Budui da se na ovaj na in mogu koristiti realniji modeli rasta, mo~e se o ekivati da e rezultati procjene prinosa biti pouzdaniji. Na taj na in mo~e se ra unati i s boljim ekonomskim efektima u poljoprivrednoj proizvodnji. Metoda procjene prinosa Pretpostavimo da su poznati podaci o kretanju prirasta neke ratarske kulture ili neke vrste stoke. Ozna imo te podatke s  QUOTE  . Ovdje  QUOTE   predstavlja vremenski trenutak,  QUOTE   veli inu prirasta, a  QUOTE   broj podataka. Podatke emo upotrijebiti u svrhu procjene parametra matemati kog modela s kojim emo opisivati prirast. Pri tome emo upotrijebiti nekoliko najpoznatijih modela sa zasienjem. OPA EKSPONENCIJALNA FUNKCIJA. Funkcija model  QUOTE   (1) koju nazivamo Opa eksponencijalna funkcija. Uz po etni uvjet  QUOTE   funkcija (1) prima oblik  QUOTE   (2) VAN BERTALANFY MODEL . Matemati ki model opisan diferencijalnom jednad~bom:  QUOTE   (3) je Bernoullijeva diferencijalna jednad~ba ije je ope rjeaenje  QUOTE   (4) gdje je  QUOTE  , pri emu je c integraciona konstanta. Uz po etni uvjet QUOTE   funkcija (4) prima oblik  QUOTE   (5) LOGISTI KA FUNKCIJA. Rjeaenja tzv. Verhulstove diferencijalne jednad~be iz 1836. Godine  QUOTE   (6) je poznata logisti ka funkcija  QUOTE   (7) U logisti kom modelu (6) pretpostavlja se da na prirast utje e trenutna vrijednost varijable  QUOTE   i bioloaki potencijal  QUOTE  . Logisti ka funkcija (7) centralno je simetri na s obzirom na to ku infleksije  QUOTE  . GENERALIZIRANA LOGISTI KA FUNKCIJA. Da bi se naglasio utjecaj trenutne vrijednosti varijable  QUOTE   ili bioloakog potencijala  QUOTE   Nelder (1961) kod bioloakih istra~ivanja i Lewandowski ko marketinakih istra~ivanja postavili su model  QUOTE   (8) Rjeaenje jednad~be (8) je tzv. generalizirana logisti ka funkcija  QUOTE   (9) Njezin graf je negativno simetri an, a to ka infleksije  QUOTE   je kod bioloakih organizama ni~e postavljena (koeficijent asimetrije  QUOTE   znatno je manji od 1). Svi navedeni modeli asimptotskog rasta imaju oblik  QUOTE  , (10) gdje je A gornja asimptota, a p vektor parametra, koje treba procjeniti na osnovi podataka mjerenja  QUOTE  . Procjena se obavlja na bazi metode najmanjih kvadrata minimiziranjem funkcionala  QUOTE   (11) ako pretpostavimo da je vektor  QUOTE   poznat onda funkcional  QUOTE   postaje funkcija jedne varijable  QUOTE   (12) gdje je  QUOTE  , a njegova prva i druga derivacija zadane su sa:  QUOTE   (13) Ovdje je lako zaklju iti da funkcija posti~e svoj globalni minimum u to ki  QUOTE   (14) Koristei (14) funkcional (11) mo~emo zapisati kao   QUOTE    Prije svega je potrebno provesti minimalizaciju funkcionala G. Ako se vektor parametar sastoji samo od jedne komponente, onda treba primjeniti algoritam za jednodimenzionalnu minimizaciju. 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Nakon toga primjenom formule (14) direktno dobivamo tra~enu vrijednost parametra A. Ovako izgleda funkcija g za navedene funkcije-modele rasta: OPA EKSPONENCIJALNA FUNKCIJA:  VAN BERTALANFY FUNKCIJA:  LOGISTI KA FUNKCIJA:  GENERALIZIRANA LOGISTI KA FUNKCIJA:  Za kraj emo navesti joa dva primjera da bismo vidjeli kako to izgleda u prakti nom slu aju. PRIMJER 1. Zadani su podaci o prirastu te~ine kokica  QUOTE   u kg tijekom prvih 10 tjedana  QUOTE   rasta 123456789100.1470.3580.6410.981.3581.7582.1592.5492.9153.251 Primjenom prethodno navedenih matemati kih modela rasta, a na osnovi danih podataka, procjenjuje se kona na te~ina kokice kao optimalna vrijednost parametra A u modelu. Pri tome se prora uni izvode s podacima po evai od 5. tjedna. Opa eksponencijalna funkcija ne mo~e se primjeniti, jer je na osnovi danih podataka prametar c negativan. Za generaliziranu logisti ku funkciju najprije se procjenjuju parametri na osnovi svih deset podataka. Tako dobiveni parametar  QUOTE   nakon toga je koriaten u svim prora unima. U slijedeoj tablici prikazane su vrijednosti procjene kona nog prirasta na bazi prvih k, k=5,6,7,8,9,10 podataka za sve promatrane matemati ke modele. PROCJENA PRIRASTA KOKICA Prora unKraj tjednaOpi eksp. modelVon Bertalanfy modelLogisti ki modelGeneralizirani logisti ki model1 2 3 4 5 65 6 7 8 9 10- - - - - -2.24 2.84 3.36 3.80 4.14 4.411.90 2.35 2.75 3.11 3.42 3.702.70 3.21 3.61 3.94 4.21 4.41 PRIMJER 2. Zadani su podaci o prirastu te~ine nema kog landrasa  QUOTE   u kg od trenutka kad je postigao 30 kg dok ne postigne 100 kg 901041181321461601741881943039485867768695100 Primjenom prethodno navedenih matemati kih modela rasta, a na osnovi danih podataka, procjenjuje se kona na te~ina ove svinjske pasmine kao optimalna vrijednost parametra A u modelu. Pri tome se prora uni izvode s podacima po evai od 5. mjerenja. Sli no kao u prethodnom primjeru opu eksponencijalnu funkciju nije mogue primjeniti, jer je na osnovi danih podataka paramtar c primio negativnu vrijednost. Za generaliziranu logisti ku funkciju primjenjen je takoer postupak kao u prethodnom primjeru. U slijedeoj tablici prikazane su vrijednosti procjene kona nog prirasta na bazi prvih k, k=5,6,7,8,9,10 podataka za sve promatrane matemati ke modele. Prora unDan Opi exp. model Von Bertalanfy modelLogisti ki modelGeneralizirani logisti ki model1 2 3 4 55 6 7 8 9- - - - -163 164 172 177 183101 107 120 129 136 140 141 161 170 183 Zaklju ak Dva bitna podru ja ekonomije, vezana uz odlu ivanje, su matemati ka ekonomija i ekonometrija. Matemati ka ekonomija koristi se za formaliziranje ekonomskih modela koje ekonomska teorija pretpostavlja dok ekonometrija primjenjuje statisti ke metode na stvarne podatke da bi ocjenila modele koje ekonomska teorija pretpostavlja. Ovim radom ukazujemo da stoji na raspolaganu mnoatvo zahtjevnih modela koji mogu biti od velike koristi u procjeni prinosa Literatura M. Crnjac, Matematika za ekonomiste, Ekonomski fakultet u Osijeku, Osijek (2001) M. Crnjac i Z. Toluai, On the estimotion of optimal parameters of some production functions, Proceedings of th 6 th International Conference on Operational Research, Rovin (1996), str. 119-124 M. Crnjac i Z. Toluai, A method for Yield estimotion, Mathematical Communication (2001). str. 139-145 R. Scitovski, Problemi najmanjih kvadrata, Financijska matematika, Ekonomski fakultet u Osijeku, Osijek (1993)  HYPERLINK "http://www.mathos.hr/~jukicd" D. Juki and  HYPERLINK "http://www.mathos.hr/~scitowsk" R. Scitovski, Least squares fitting Gaussian type curve, Appl. Math. Comput. 167 (2005), 286-298. G.Kralik, R. Scitovski, Istra~ivanje zna ajki rasta brojlera pomou asimetri ne S-funkcije, Sto arstvo 47(1993), 207-213. G.Kralik, R. Scitovski, D.Sen i, Application of asymmetric S-function for analysis and valuation of the growth of boars, Sto arstvo 47(1993), 425-433. R.Scitovski, G.Kralik, K.Sabo, T.Jelen, A mathematical model of controlling the growth of tissue in pigs, Appl. Math. 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