Stochastic Programming Model in Agricultural Production Decision Support System

Transcription

1 Stochastc Programmg Model Agrcultural Producto Decso Support System Gordaa Duk ABACUS Tuto, Research ad Busess Cosultacy Mosorska 8, 1000 Osek, Croata Zdravko Toluš Departmet of Agroecoomcs, Faculty of Agrculture Uversty of Josp Jura Strossmayer Osek Trg Sv. Trostva, Osek, Croata Zdravko.Tolusc@pfos.hr Mate Sesar Ph.D. Studet - Faculty of Agrculture Uversty of Josp Jura Strossmayer Osek Trg Sv. Trostva, Osek, Croata matesesar@et.hr Abstract. Agrcultural producto depeds o varous factors whch are frequetly mpossble to cotrol to a suffcet degree. Cosequetly, t s ot possble to determe precsely parameter values, whch s a maor problem for decso-makers whe puttg up a pla for the type ad quatty of agrcultural output that should be produced ad offered o the market. Sce applyg determstc models caot yeld satsfactory results such crcumstaces, t s ecessary to mprove the decso support system by mplemetg stochastc programmg. Its basc feature s defg parameters of the model as radom varables. I order to establsh solutos to the problem set ths way, the stochastc programmg model has to be trasformed to a determstc form, whch the results oe or more costrats of o-lear character. Gve that ths form of mathematcal optmzato s very comple to solve, t requres maagers to use computers ad adequate software order to carry t out. I ths paper, the proposed model of decsomakg agrcultural producto, based o computer optmzato, s frst cosdered from a theoretcal vewpot. After that, the procedure of ts costructo ad soluto s llustrated o a hypothetcal eample. Keywords. Agrcultural producto plag, stochastc programmg model, decso support system, computer optmzato 1 Itroducto Agrcultural producto s characterzed to a great degree by ucertaty, whch makes plag more dffcult. Apart from the huma ad techcal factors, agrcultural polcy measures ad ucerta codto o the agrcultural products market, ths producto s greatly flueced by dfferet atural factors. If coducted due tme, there are actvtes that ca allevate egatve mpacts of some atural factors. Thus, for eample, vestmets to rrgato ad draage systems ca help us to avod serous damage caused by drought or ecessve rafall, whereas crop rotato plag ad adequate fertlzato ca mprove sol qualty. Gve all that, there stll remas a varety of factors agrcultural producto that are ot fully predctable,.e. they are susceptble to chage. It follows that the parameters represetg these factors caot be determed wth certaty. I such codtos, applyg determstc models agrcultural output plag s ot ustfed. I order to mprove decso-makg ths doma, ths paper proposes a stochastc programmg model based o computer optmzato. The decso support system

2 mproved ths way becomes sutable for usage codtos of ucertaty. Because of ts character, moder formato techologes play a mportat role ths model. Research methodology The paper presets theoretcally ad o a smple hypothetcal eample the process of costructg ad solvg the decso makg model agrcultural producto, whch s fouded o stochastc programmg. Stochastc programmg models assume that parameters are ot kow wth certaty, rather, they are defed as radom varables that follow oe of theoretcal dstrbutos. I the proposed model t s assumed that radom varables are ormal dstrbuted, ad that at least oe costrat s stochastcally set. Whe solvg the problem posed ths way, t s ecessary to establsh determstc equvalets of all probablstc costrats. I ths way, we get to the o-lear costrats. Sce solvg ths kd of a problem s hghly comple, t calls for adequate computer support. I ths paper the program package Mathematca was used. The soluto of the posed problem ultmately determes the quatty of agrcultural output whch should be produced ad offered to the market, uder gve lmts. Model formulato Determstc models assume that the parameters are kow wth certaty, whch makes t possble to establsh optmal solutos to the posed problems. I cotrast, stochastc models treat ther parameters as radom varables, whch makes t ecessary to forecast the realzato probablty of the costrats whch they appear. A geeral formulato for stochastc programmg models defed ths way, case whe the obectve fucto s mamzed, reads as follows: It s thus assumed the model that parameters a * ( = 1,,m, = 1,,) ad rghthad sde value of costrat b * ( = 1,,m) ca be defed as radom varables. The costrats that do ot cota such parameters are treated as determstc ths model. Wth regard to parameters defed as radom varables, whe establshg determstc equvalets of probablstc costrats we eed to dfferetate betwee three cases. I the frst case t s assumed that parameters a * are defed as radom varables that follow ormal dstrbuto, wth mea ad varace. A determstc form of the th costrat, stochastcally posed ths way, reads as follows: K 1 1 b It should be oted that the stated epresso K represets a cumulatve dstrbuto fucto of stadard ormal dstrbuto. I the secod case, t s assumed that the rght-had sde value of costrat b * s defed as radom varable that follow ormal dstrbuto, wth mea ad varace. A determstc equvalet of thus posed th costrat s the followg: a 1 K The thrd case appears whe both parameters a * ad the rght-had sde value of costrat b * are defed as radom varables th costrat. Here t s also assumed that parameters a * are ormal dstrbuted, wth mea ad varace, ad that the rght-had sde value of costrat s ormal dstrbuted as well, wth mea ad varace. A determstc form of the stochastc costrat posed ths way ow reads as follows: mamze z c 1 K 1 1 subect to P * * a b 1, 1,..., m 1 a 1 b, m 1,..., r 0, 1,..., If ths proves ecessary, whe posg the stochastc programmg model a decso-maker ca combe dfferet forms of probablstc costrats. The basc steps costructg, solvg ad mplemetg the aalyzed model are show Fgure 1.

3 DEFINING THE PROBLEM AND VARIABLES SPECIFICATION MODELLING AND CONSTRUCTION OF AN ADEQUATE DATABASE STOCHASTIC PROGRAMMING MODEL FORMULATION DEFINING THE DETERMINISTIC EQUIVALENT OF THE SET MODEL DETERMINING THE SOLUTION BY MEANS OF COMPUTER SYSTEM NO THEORETICAL AND PRACTICAL PREREQUISITES FOR MODEL APPLICATION ARE MET YES MODEL IMPLEMENTATION WITH CONTINUOUS VERIFICATION OF ADEQUACY Fgure 1. The basc steps costructg, solvg ad mplemetg the aalyzed model I the frst phase of model costructo, t s ecessary to defe the problem ad specfy all relevat model varables. Eperece of eperts who have bee volved process of agrcultural producto ca be of great help to maagers ths step. Thers eperece eable maagers to observe more precsely all the elemets of aalyzed problem. Modellg ad costructo of a adequate database s the et phase model buldg. Ths database should cota dfferet data whch s relevat for solvg the problem. Some of them are data that refers to crops produced durg the passed years, epected costs ad comes, ad the avalable quatty of producto materals. A moder formato ad commucato techology allows maager to create, model, establsh ad use such database relatvely smply way. After determg all relevat varables ad thers parameters, t s possble to formulate the stochastc programmg model. As prevously metoed, such a model some costrats ca be defed as determstc, but at least oe has to be defed as stochastc. For all probablstc costrats we eed to defe ther determstc equvalets. Solvg ths kd of a problem wthout a adequate computer support s qute comple. It s the reaso the program package Mathematca was used ths paper. Before mplemetg the model, we eed to check whether the theoretcal ad practcal prerequstes for model applcato have bee met. If the formulated model turs out to be adequate, we have to revew each phase model costructo, whch also cludes coductg the requred modfcatos. I the forthcomg future the adequacy of the model should be cotuously verfed. 4 Numercal eample Let t be assumed that the maagemet of a agrcultural compay wshes to determe the area, epressed hectares (ha), whch should be plated wth peppers ( 1 ), lettuce ( ) ad tomatoes ( ), ad that the sellg prce has bee predetermed wth a wholesaler. Based o ths prce t s epected that the total come, cludg the cetves, wll be EUR/ha for peppers, EUR/ha for lettuce, ad 0000 EUR/ha for tomatoes. Because of volatle market codtos, producto costs caot be determed wth certaty, rather, they are defed the model as radom varables that follow ormal dstrbuto. It s thus estmated that the epected producto costs for pepper wll be EUR ( 11 ) wth varace EUR ( 11), for lettuce EUR ( 1 ) wth varace 0000 EUR ( 1), ad for tomatoes EUR ( 1 ) wth varace 0000 EUR ( 1). The total amout at dsposal for producg the vegetables uder cosderato here s EUR. Amog other thgs, sulphate fertlzer NPK 5:0:0 s used to fertlze peppers, lettuce ad tomatoes. Peppers requre 600 kg/ha of that fertlzer, lettuce requres 5 kg/ha, ad tomatoes requre 500 kg/ha. I our eample, t wll be assumed that there are dffcultes the supply of ths fertlzer due to producto problems. For ths reaso, the parameter represetg the

4 quatty at dsposal wll be defed as radom varable that follows ormal dstrbuto, wth mea 9500 kg ( ) ad varace 8100 kg ( ). The agrcultural compay plas to cultvate o more tha 5 ha wth peppers, lettuce ad tomatoes. To meet the cotractual oblgatos to the wholesaler, peppers must be plated o 5 ha at least, whereas the area uder tomatoes ca be mamally twce as large as the oe uder lettuce. It wll be assumed that the frst stochastc costrat wll be realzed wth a mmum probablty of 90%, ad the secod wth a mmum probablty of 95%. The stochastc model of the problem thus defed, wth the obectve to mamze total come, reads as follows: mamze subect to z * * * 11 1 a1 a P b * 0 95 P a , 0, The determstc equvalet of the set model s the followg: 0 mamze subect to z S S , 0, Determg a soluto to the resultg model s cocevable wthout computer systems. Ther usage s oe of the basc features of decso support systems, because they corporate formato techology order to advace the process of solvg the problem. I ths way computer systems mprove the decsomakg process. For ths purpose ths partcular case we used program package Mathematca. Ths applcato allows us to easly solve a dfferet kd of olear optmzato problem. Fgure shows Mathematca s oscree dsplay of aalyzed problem. The program package Mathematca has yelded the followg solutos: 1 =5, = ad = Wth these areas uder cultvato, t s epected that the agrcultural compay wll acheve the come of 467 EUR. Fgure. Mathematca s oscree dsplay of aalyzed problem

5 5 Coclusos Approprately mplemeted stochastc programmg models, whch are teded for use codtos of ucertaty, ca be a powerful tool for agrcultural maagemet producto plag. Icludg such models to a decso support system that deals wth the type ad quatty of agrcultural output to be produced ad offered to the market ca mprove the whole maagemet process. I addto to the adequate quattatve model, decso support systems corporate also the formato techology requred for solvg, aalyss ad modfcato of the set model. Wthout computer support t would be vrtually mpossble to solve a problem of o-lear optmzato, obtaed after establshg determstc forms of stochastcally set costrats. Moder program applcatos are so easy to use that eve the maagers who are o computer eperts ca apply the proposed stochastc programmg model wthout much dffculty. Refereces [1] Aderso, D.R., Sweeey, D.J., Wllams, T.A.: A Itroducto to Maagemet Scece: Quattatve Approaches to Decso Makg, 10th Edto, West Publshg Co., St. Paul, 00. [] Bogle, I.D.L., Žlskas, J. (eds.): Computer Aded Methods Optmal Desg ad Operatos (Seres o Computers ad Operatos Research), World Scetfc Publshg Compay, Sgapore, 006. [] Darby-Dowma, K., Barker S., Audsley E., Parsos D.: A Two-Stage Stochastc Programmg wth Recourse Model for Determg Robust Platg Plas Hortculture, Joural of the Operatoal Research Socety, Palgrave Macmlla Ltd., Basgstoke, 000, Volume 51, Number 1, pp [4] Kall, P., Wallace, S.W.: Stochastc Programmg, Joh Wley & Sos, Chchester, [5] Koutsouks, N.S., Mtra, G.: Decso Modellg ad Iformato Systems: The Iformato Value Cha (Operatos Research/Computer Scece Iterfaces Seres), Kluwer Academc Publshers, Norwell, 00. [6] Peart, R.M., Curry, R.B. (eds.): Agrcultural Systems Modelg ad Smulato, Marcel Dekker, Ic., New York, [7] Scheder, J.J., Krkpatrck, S.: Stochastc Optmzato (Scetfc Computato), Sprger, Berl, 006. [8] Shvely, G.E.: Sol Coservato ad Cosumpto Rsk a Model of Low-Icome Agrculture, Evrometal Motorg ad Assessmet, Sprger, New York, 000, Volume 6, Number 1, pp [9] Taha, H.A.: Operatos Research - A Itroducto, Seveth Edto. Pretce-Hall, New York, 00. [10]Wllams, H.P.: Model Buldg Mathematcal Programmg, Fourth Edto, Joh Wley & Sos Ltd., Chchester, [11]Wsto, W.L.: Operatos Research - Applcatos ad Algorthms. Fourth Edto. Brooks/Cole-Thomso Learg, Belmot, 004.