An Unconstrained Q - G Programming Problem and its Application

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Joual of Ifomato Egeeg ad Applcatos ISS 4-578 (pt) ISS 5-0506 (ole) Vol.5, o., 05 www.ste.og A Ucostaed Q - G Pogammg Poblem ad ts Applcato M. He Dosh D. Chag Tved.

1 Joual of Ifomato Egeeg ad Applcatos ISS (pt) ISS (ole) Vol.5, o., 05 A Ucostaed Q - G Pogammg Poblem ad ts Applcato M. He Dosh D. Chag Tved.Assocate Pofesso, H L College of Commece, avagua, Ahmedabad Head & Assocate Pofesso, R J T Commece College, Vastapu, Ahmedabad E-mal of the coespodg autho: chagtved580@ahoo.com Abstact I pactce thee est ma methods to solve ucostaed, costaed ad med quadatc ad geometc pogammg poblems. I ths pape a attempt s made to develop ucostat Q G pogammg poblem b combg quadatc pogammg poblem ad geometc pogammg poblem. Ths model s solved usg the techque of geometc pogammg poblem. A hpothetcal eample s cosdeed to llustate the model. Kewods: Ucostaed Quadatc Pogammg poblems, Geometc Pogammg poblem, Pmal ad Dual poblem, Othogoalt ad omalt codtos.. Itoducto I pactce vaous techques ae avalable to solve ucostaed quadatc poblem. The objectve of quadatc pogammg s to mame o to mme the quadatc objectve fucto. Let decso vaables ad the C R ad D be smmetc mat of eal umbes of ode the the coeffcets of objectve fucto, ucostaed quadatc pogammg poblem s defe as follows: ' ' Mme f ( ) C + D ' D D ( d j) Hee s quadatc fom ad s a smmetc mat. Geometc pogammg s a techque fo solvg a specal case of olea poblems. Duff, Peteso ad Zee [] publshed a boo Geometc Pogammg: Theo ad Applcatos that stated the feld of Geometc Pogammg as a bach of olea optmato wth ma useful theoetcal ad computatoal popetes of Geometc Pogammg, to a lage etet the scope of lea pogammg applcatos ad s atuall appled to seveal mpotat olea sstems scece ad egeeg. Seveal mpotat developmets of Geometc Pogammg ae the aea of mechacal ad cvl egeeg, chemcal egeeg, pobablt ad statstcs, face ad ecoomcs, cotol theo, ccut desg, fomato techolog, codg ad sgal pocessg, weless etwog, etc. too place the late 960s to eal 970s. Thee ae seveal boos o olea optmato that have a secto o Geometc Pogammg, e.g., M. Avel, [5], C. S. Beghtle [], G. Hadle [4], Taha [6], etc. Howeve, ma eseaches felt that most of the theoetcal, algothmc ad applcato aspects of Geometc Pogammg had bee ehausted b the eal 980 s, the peod of was elatvel quet. Afte the evoluto the electoc feld, ove the last few eas, Geometc Pogammg stated to eceve eewed atteto fom the opeatos eseach commut. I 964, R. Duff ad C. Zee [], have defed ucostaed Geometc Pogammg the followg mae: Z f ( ) Whee, U a U C fo,,,..., c > 0 a ad s fte. The epoets ae uestcted sg.e. t Hee t s assumed that the coeffcet ma be postve o egatve. A model s cosdeed whch the cocept of quadatc pogammg poblem ad geometc pogammg s combed, s defed as Q G model..assumuptos I the peset stud followg assumptos ae made to deve a soluto to the quadatc-geometc [] 50

2 Joual of Ifomato Egeeg ad Applcatos ISS (pt) ISS (ole) Vol.5, o., 05 pogammg. c 0 0 ;,,.... The coeffcets ae uestcted sg.e. o c. s fte.e. umbe of tems s fte.. The umbe of tems + whee s umbe of vaables. D ( d j) 4. s a smmetc mat.. Mathematcal Model ad Pocedue: Let ad C R ad D s a eal ' ' Mmse Z f C D mat the ucostat Q - G pogammg poblem s ( ) + c d j + j j defed as ude: The above poblem ca be coveted to geometc pogammg poblem as ude: Whee, ( ) Z f U a U c fo,,,..., Ths poblem wll be cosdeed as the pmal poblem. Hee Z s the polomal fom ad t s assumed that all vaables of the esults. 0 ae stctl postve o egatve. The equemet [] plas a essetal ole the devato Fo mmum value of the objectve fucto, the fst ode patal devatve of Z must be eo, ow dffeetate Z wth espect to U Sce, each ;,,... ' a a ( ) 0,,,,,..., c a fo 0 0 au, fo,,,...,,,,..., Let be the mmum value of Z It s ecessa that defed as U 0 Whch shows that ad Q 0, sce each [] [4] 0 ad Z s a polomal th Thus the value of epesets the elatve combato of the U tem to the optmum value of the [5] [6] 5

3 Joual of Ifomato Egeeg ad Applcatos ISS (pt) ISS (ole) Vol.5, o., 05 objectve fucto. ow the ecessa codtos ca be wtte as ad a 0 ( > 0, 0,,,... ), ( 0,,,,... ) [8] These codtos [7] ad [8] ae ow as othogoalt ad omalt codtos. B usg mat veso method, these codtos wll gve a uque soluto fo, f (+) ad all the equatos ae depedet. If > (+) the the poblem becomes moe comple because the values of ae ot uque. Howeve, t s possble to deteme uquel fo the pupose of mmg Z. ow, suppose that ae the uque values detemed fom the equatos gve the esults [7] ad [8]. These values ae used to detemed the values of ad fo,,,... as ude, Cosde, ( ) Q C a C a a C C Q a 0 Thus, the value of s detemed fom esult [] as soo as all ae detemed. ow, fo ow values of ad the value of U ca be detemed fom U Sce a ( ) U C fo,,,..., fo,,,.... [7] [9] [0] [] [] [] smultaeousl soluto of these equatos should gve The pocedue descbed hae shows that the soluto to the ogal polomal Z ca be tasfomed to the soluto of a set of lea equatos. Obseved that all ae detemed fom the ecessa codtos fo a mmum. Howeve, t ca be show that, these codtos ae also suffcet. 5

4 Joual of Ifomato Egeeg ad Applcatos ISS (pt) ISS (ole) Vol.5, o., Cocluso Hpothetcal Poblem: Cosde the followg poblem of Q - G Pogammg wth thee decso vaables ad fou tems; 5 M m se Z + The above poblem ca be wtte as Mmse Z + 5 U 0 0 +U +U + + a a a a a a c c c The b compaso, followg matces ae obtaed. ( c c c ) ( ) 5 a a a 0 a a a 0 ad Hee, ad,, so the case whch (+) s to be cosdeed. Usg othogoalt ad omalt codtos a 0 ad Followg equatos ae obtaed a + a + a 0 a + a + a 0 ad The above equatos ca be epeseted as, ow, b usg mat veso method the values of dual vaables ca be detemed as ude, Ths s a uque soluto gve as,, ow, c

5 Joual of Ifomato Egeeg ad Applcatos ISS (pt) ISS (ole) Vol.5, o., 05 u Fom the equato t ca be deduced that, u 0.96, u 0.48, u 0.96 Whch wll gve the optmum soluto to the pmal poblem as ude:.08,.5 ad Refeeces C. S. Beghtle ad D. T. Phlps, Appled Geometc Pogammg. Wle, 976. R. J. Duff, Leaed geometc pogams, SIAM Revew, vol., pp. 7, 970. R. J. Duff, E. L. Peteso, ad C. Zee, Geometc Pogammg: Theo ad Applcatos, Wle, 967. G Hadle : olea ad Damc Pogammg Addso Wesle Publshg compa, 964. M. Avel, M. J. Rjcaet, ad D. J. Wlde, Optmato ad Desg. Petce Hall, 97. H A Taha, Opeato Reseach Itoducto Petce Hall of Ida Pvate Lmted,

The IISTE s a poee the Ope-Access hostg sevce ad academc evet maagemet. The am of the fm s Acceleatg Global Kowledge Shag. Moe fomato about the fm ca be foud o the homepage: http://www.ste.

6 The IISTE s a poee the Ope-Access hostg sevce ad academc evet maagemet. The am of the fm s Acceleatg Global Kowledge Shag. Moe fomato about the fm ca be foud o the homepage: CALL FOR JOURAL PAPERS Thee ae moe tha 0 pee-evewed academc jouals hosted ude the hostg platfom. Pospectve authos of jouals ca fd the submsso stucto o the followg page: All the jouals atcles ae avalable ole to the eades all ove the wold wthout facal, legal, o techcal baes othe tha those sepaable fom gag access to the teet tself. Pape veso of the jouals s also avalable upo equest of eades ad authos. MORE RESOURCES Boo publcato fomato: Academc cofeece: IISTE Kowledge Shag Pates EBSCO, Ide Copecus, Ulch's Peodcals Decto, JoualTOCS, PKP Ope Achves Haveste, Belefeld Academc Seach Ege, Eletosche Zetschftebblothe EZB, Ope J-Gate, OCLC WoldCat, Uvese Dgtal Lba, ewjou, Google Schola

such that for 1 From the definition of the k-fibonacci numbers, the firsts of them are presented in Table 1. Table 1: First k-fibonacci numbers F 1

such that for 1 From the definition of the k-fibonacci numbers, the firsts of them are presented in Table 1. Table 1: First k-fibonacci numbers F 1 Scholas Joual of Egeeg ad Techology (SJET) Sch. J. Eg. Tech. 0; (C):669-67 Scholas Academc ad Scetfc Publshe (A Iteatoal Publshe fo Academc ad Scetfc Resouces) www.saspublshe.com ISSN -X (Ole) ISSN 7-9