World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:10, No:9, 2016
6ROYLQJ)X]]\0XOWL2EMHFWLYH/LQHDU3URJUDPPLQJ
3UREOHPVZLWK)X]]\'HFLVLRQ9DULDEOHV
0DKQD]+RVVHLQ]DGHK$OL\HK.D]HPL
IRUPXODWLRQRI)02/33ZLWKIX]]\ULJKWKDQGVLGHDQGIX]]\
GHFLVLRQYDULDEOHVDUHGLVFXVVHGLQ6HFWLRQ,9DQHZPHWKRG
LV SURSRVHG IRU VROYLQJ )02/33SUHVHQWHG LQ 6HFWLRQ ,,, WR
LOOXVWUDWH WKH SURSRVHG PHWKRG LW LV DSSOLHG WR D UHDO OLIH
DOORFDWLRQ SUREOHP DQG WKH REWDLQHG UHVXOWV DUH GLVFXVVHG LQ
6HFWLRQ9FRQFOXVLRQLVGUDZQLQ6HFWLRQ9,
International Science Index, Mathematical and Computational Sciences Vol:10, No:9, 2016 waset.org/Publication/10005261
Abstract²,QWKLVSDSHUDPHWKRGLVSURSRVHGIRUVROYLQJ)X]]\
0XOWL2EMHFWLYH /LQHDU 3URJUDPPLQJ SUREOHPV )02/33 ZLWK
IX]]\ ULJKW KDQG VLGH DQG IX]]\ GHFLVLRQ YDULDEOHV 7R LOOXVWUDWH WKH
SURSRVHG PHWKRG LW LV DSSOLHG WR WKH SUREOHP RI VHOHFWLQJ VXSSOLHUV
IRUDQDXWRPRWLYHSDUWVSURGXFHUFRPSDQ\LQ,UDQLQRUGHUWRILQGWKH
QXPEHU RI RSWLPDO RUGHUV DOORFDWHG WR HDFK VXSSOLHU FRQVLGHULQJ WKH
FRQIOLFWLQJREMHFWLYHV)LQDOO\WKHREWDLQHGUHVXOWVDUHGLVFXVVHG
Keywords²)X]]\PXOWLREMHFWLYHOLQHDUSURJUDPPLQJSUREOHPV
WULDQJXODUIX]]\QXPEHUVIX]]\UDQNLQJVXSSOLHUVHOHFWLRQSUREOHP ,
,,35(/,0,1$5,(6
'HILQLWLRQ$IX]]\QXPEHU ‫ܣ‬ሚ ൌ ሺ݉஺ ǡ ‫ݓ‬஺ ǡ ‫ݓ‬஺ƴ ሻLVD/5W\SHLI
DQGRQO\LI
, ,1752'8&7,21
௠ ି௫
‫ ܮ‬ቀ ௪ಲ ቁ െ λ ൏ ‫ ݔ‬൑ ݉஺
ಲ
‫ݓ‬஺ ǡ ‫ݓ‬஺ƴ ൒ Ͳ
‫ܣ‬ሺ‫ݔ‬ሻ ൌ ቐ
௫ି௠
ܴ ቀ ௪ƴ ಲ ቁ݉஺ ൑ ‫ ݔ‬൏ ൅λ
1RSWLPL]LQJUHDOZRUOGV\VWHPVRQHXVXDOO\HQGVXSZLWKD
OLQHDURUQRQOLQHDUSURJUDPPLQJSUREOHP >@7KHUHDOOLIH
GHFLVLRQ SUREOHPV KDYH WZR PDLQ SURSHUWLHV 7KH ILUVW RQH LV
WRKDYHFRQIOLFWLQJREMHFWLYHVLQWKHSUREOHPVWUXFWXUHDQGWKH
VHFRQG RQH LV WKH IX]]LQHVV LQ WKH GHVFULSWLRQ RI SUREOHP
SDUDPHWHUV >@ 7KH ILUVW DSSOLFDWLRQ RI )X]]\ 6HWV 7KHRU\
)67WRGHFLVLRQPDNLQJSURFHVVHVZDVSUHVHQWHGE\%HOOPDQ
DQG =DGHK >@ DQG WKH FRQFHSW RI IX]]\ PDWKHPDWLFDO
SURJUDPPLQJRQJHQHUDOOHYHOZDVILUVWSURSRVHGE\7DQDNDHW
DO >@ 7KH ILUVW IRUPXODWLRQ RI IX]]\ OLQHDU SURJUDPPLQJ
)/3 LV SURSRVHG E\ =LPPHUPDQQ >@ 9DULRXV IX]]\ OLQHDU
SURJUDPPLQJ WHFKQLTXHV DUH VXUYH\HG LQ OLWHUDWXUH ZKLFK DUH
FODVVLILHG LQWR WZR PDLQ FODVVHV IX]]\ OLQHDU SURJUDPPLQJ
>@>@ DQG SRVVLELOLVWLF OLQHDU SURJUDPPLQJ >@>@ )RU
WKHVDNHRIVLPSOLFLW\XVXDOO\WKHVHWHFKQLTXHVFRQVLGHURQO\
FULVS VROXWLRQV RI WKH IX]]\ SUREOHPV >@ *HQHUDOO\
VSHDNLQJ LQ IX]]\ OLQHDU SURJUDPPLQJ SUREOHPV WKH
FRHIILFLHQWV RI GHFLVLRQ YDULDEOHV DUH IX]]\ QXPEHUV ZKLOH
GHFLVLRQ YDULDEOHV DUH WKH FULVS RQHV 7KLV PHDQV WKDW LQ DQ
XQFHUWDLQHQYLURQPHQWDFULVSGHFLVLRQLVPDGHWRPHHWVRPH
GHFLVLRQ FULWHULD >@ WKXV WKH GHFLVLRQPDNLQJ SURFHVV LV
FRQVWUDLQHGWRFULVSWKHGHFLVLRQVWKDWKLGHWKHIX]]\DVSHFWRI
WKH SUREOHP >@ 7KHQ VRPH DXWKRUV KDYH GHYHORSHG WKH
IX]]\ OLQHDU SURJUDPPLQJ SUREOHPV LQ ZKLFK DOO WKH
SDUDPHWHUV DV ZHOO DV WKH YDULDEOHV DUH UHSUHVHQWHG E\ IX]]\
QXPEHUV LV NQRZQ DV ))/3 SUREOHPV >@>@ 7KHVH
SUREOHPVGRQRWVKRZHQRXJKXQGHUVWDQGLQJRIWKHH[WHQVLRQ
SULQFLSOH
7KHSDSHULVRUJDQL]HGDVIROORZVLQ6HFWLRQ,,VRPHEDVLF
GHILQLWLRQV RI IX]]\ VHW WKHRU\ DQG DULWKPHWLF EHWZHHQ WZR
WULDQJXODU IX]]\ QXPEHUV DUH UHYLHZHG LQ 6HFWLRQ ,,,
ಲ
ZKHUH݉஺ LVWKHFHQWHUDQG‫ݓ‬஺ DQG‫ݓ‬஺ƴ DUHWKHOHIWDQGULJKW
EDQGZLGWKV RI ‫ܣ‬ሚ UHVSHFWLYHO\ 7KLV LV D SDUDPHWULF IRUP RI
IX]]\ QXPEHU ‫ܣ‬ሚ VR ZH FDQ VKRZ LW DV D WULDQJXODU VKDSH DV
‫ܣ‬ሚ ‫ ؠ‬ሺ݉ǡ ‫ݓ‬஺ ǡ ‫ݓ‬ƴ ஺ ሻ௅ோ 'HILQLWLRQ *LYHQ WZR WULDQJXODU IX]]\ QXPEHUV ‫ܣ‬ሚ ൌ
෨ ൏ ‫ܣ‬ሚLIDQGRQO\LI
ሺܽǡ ‫ݓ‬஺ ǡ ‫ݓ‬ƴ ஺ ሻ௅ோ DQG‫ܤ‬෨ ൌ ሺܾǡ ‫ݓ‬஻ ǡ ‫ݓ‬ƴ ࡮ ሻ௅ோ ‫ܤ‬
EDDQG‫ݓ‬஻ ൅‫ݓ‬ሖ࡮ ൒ ‫ݓ‬஺ ൅‫ݓ‬ƴ ஺ DQG‫ܤ‬෨ ൌ ‫ܣ‬ሚLIDQGRQO\LI E
DDQG ‫ݓ‬஻ ൅‫࡮ݓ‬ƴ ൌ ‫ݓ‬஺ ൅‫ݓ‬ƴ ஺ 'HILQLWLRQ$WULDQJXODUIX]]\QXPEHU ‫ܣ‬ሚ ൌ ሺܽǡ ‫ݓ‬஺ ǡ ‫ݓ‬ƴ ஺ ሻ௅ோ LV
VDLGWREHQRQQHJDWLYHIX]]\QXPEHULIDQGRQO\LI ܽ െ ‫ݓ‬஺ ൒
Ͳ
'HILQLWLRQ *LYHQ WZR IX]]\ QXPEHUV ‫ܣ‬ሚ ൌ ሺܽǡ ‫ݓ‬஺ ǡ ‫ݓ‬ƴ ஺ ሻ௅ோ
DQG ‫ܤ‬෨ ൌ ሺܾǡ ‫ݓ‬஻ ǡ ‫ݓ‬ƴ ࡮ ሻ௅ோ ZLWK FRQWLQXRXV QRQGHFUHDVLQJ
IXQFWLRQ RYHUሾͲǡ λሻ IX]]\ $ULWKPHWLF RSHUDWLRQV DUH GHILQHG
DV
ሖ ‫࡮ݓ‬ƴ ൯ ‫ ܣ‬൅ ‫ ܤ‬ൌ ൫ܽ ൅ ܾǡ ‫ݓ‬஺ ൅ ‫ݓ‬஻ ǡ ‫ݓ‬஺ ൅
௅ோ
‫ ܣ‬െ ‫ ܤ‬ൌ ሺܽ െ ܾǡ ‫ݓ‬஺ ൅ ‫࡮ݓ‬ƴ ǡ ‫ݓ‬஺ ൅ƴ ‫ ࡮ݓ‬ሻ௅ோ ሺܽǡ ‫ݓ‬஺ ǡ ‫ݓ‬ƴ ஺ ሻ௅ோ Ǥ ሺܾǡ ‫ݓ‬஻ ǡ ‫ݓ‬ƴ ࡮ ሻ௅ோ ൎ ሺܽǤ ܾǡ ܽ‫ ࡮ݓ‬൅ ܾ‫ݓ‬஺ ǡ ܽ‫࡮ݓ‬ƴ ൅ ܾ‫ݓ‬஺ƴ ሻ௅ோ 0DKQD] +RVVHLQ]DGHK LV ZLWK WKH 'HSDUWPHQW RI ,QGXVWULDO 0DQDJHPHQW
)DFXOW\ RI 0DQDJHPHQW 8QLYHUVLW\ RI 7HKUDQ 7HKUDQ ,UDQ FRUUHVSRQGLQJ
DXWKRUSKRQHHPDLOPKRVVHLQ]DGHK#XWDFLU
$OL\HK.D]HPLLVZLWKWKH'HSDUWPHQWRI,QGXVWULDO0DQDJHPHQW)DFXOW\
RI 0DQDJHPHQW 8QLYHUVLW\ RI 7HKUDQ 7HKUDQ ,UDQ HPDLO
DOL\HKND]HPL#XWDFLU
ሺܽǡ ‫ݓ‬஺ ǡ ‫ݓ‬ƴ ஺ ሻ௅ோ Ǥ ሺܾǡ ‫ݓ‬஻ ǡ ‫ݓ‬ƴ ࡮ ሻ௅ோ ൎ ሺܽǤ ܾǡ െܽ‫࡮ݓ‬ƴ ൅ ܾ‫ݓ‬஺ ǡ െܽ‫ ࡮ݓ‬൅ ܾ‫ݓ‬஺ƴ ሻ௅ோ ܿǤ ሺܽǡ ‫ݓ‬஺ ǡ ‫ݓ‬ƴ ஺ ሻ௅ோ ൌ ሺܿܽǡ ܿ‫ݓ‬஺ ǡ ܿ‫ݓ‬ƴ ஺ ሻ௅ோ International Scholarly and Scientific Research & Innovation 10(9) 2016
417
scholar.waset.org/1999.7/10005261
World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:10, No:9, 2016
,,, $08/7,2%-(&7,9(/,1($5352*5$00,1*352%/(0
:,7+75,$1*8/$5)8==<9$5,$%/(6$1'5,*+7+$1'6,'(
180%(5
6WHS'HILQHDQHZVLQJOHREMHFWLYHIXQFWLRQVXEMHFWWRFULVS
FRQVWUDLQWVGHILQHGLQ6WHSDV
$ PXOWL REMHFWLYH OLQHDU SURJUDPPLQJ SUREOHP ZLWK
WULDQJXODU IX]]\ YDULDEOHV DQG ULJKW KDQG VLGH QXPEHU LV
GHILQHGDV
International Science Index, Mathematical and Computational Sciences Vol:10, No:9, 2016 waset.org/Publication/10005261
ஸ
‫ݏ‬Ǥ ‫ݐ‬Ǥ σ௡௝ୀଵ ܽ௜௝ ‫ݔ‬௝௠ ൬ୀஹ൰ ܾ௜௠
ZKHUH ™୔ ǡ ൌ ͳǡ ǥ ǡ ’ LV WKH ZHLJKW RI REMHFWLYH IXQFWLRQ 3
GHWHUPLQHGE\'0
6WHS)LQG WKH RSWLPDO VROXWLRQ RI 6WHS ZKLFK LV DQ
HIILFLHQWVROXWLRQRIWKHRULJLQDO)02/3SUREOHP
‫ݔ‬෤ ൌ ሺ‫ݔ‬௠ ǡ ‫ݓ‬ǡ ‫ݓ‬ƴሻǡ ܾ ൌ ሺܾǡ ܾ௪ ǡ ܾ௪ƴ ሻǡ ‫ݔ‬௠ ൌ ܿ‫݁ݎ݋‬ሺ‫ݔ‬෤ሻǡ ܾ ൌ ܿ‫݁ݎ݋‬൫ܾ෨൯
9$33/,&$7,212)7+(352326('02'(/727+(6833/,(5
6(/(&7,21352%/(0
DQG ‫ݓ‬ǡ ‫ݓ‬ƴǡ ܾ௪ ǡ ܾ௪ƴ DUH WKH EDQGZLGWKV RI ‫ݔ‬෤ DQG ܾ෨ UHVSHFWLYHO\
‫ݔ‬෤߳‫ ܧ‬௡ ǡ ܾ෨߳‫ ܧ‬ெ ǡ ‫ ܣ‬ൌ ሾܽ௜௝ ሿெൈ௡ ܴ߳ ெൈ௡ ǡ ‫ܴ߳ܥ‬௡ 'HILQLWLRQ :H VD\ WKDW IX]]\ YHFWRU ‫ݔ‬෤ ൌ ሺ‫ݔ‬௝௠ ǡ ‫ݓ‬௝ ǡ ‫ݓ‬ƴ௝ ሻLV D
IX]]\ IHDVLEOH VROXWLRQ WR WKH SUREOHP LI ‫ݔ‬෤ VDWLVILHV WKH
FRQVWUDLQWVRIWKHSUREOHP
'HILQLWLRQ$ IX]]\ IHDVLEOH VROXWLRQ‫ݔ‬෤ ‫ כ‬LV D IX]]\RSWLPDO
VROXWLRQRILIIRUDOOIX]]\IHDVLEOHVROXWLRQš෤IRUZH
෪ሺ൑
෩ ሻ…š෤
KDYH…š෤ ‫ כ‬൒
,Q WKLV VHFWLRQ WKH SURSRVHG DSSURDFK LV DSSOLHG WR D
VXSSOLHU VHOHFWLRQ DQG RUGHU DOORFDWLRQ SUREOHP IRU DQ
DXWRPRWLYHSDUWSURGXFHUFRPSDQ\
7KHFRPSDQ\QHHGVDERXWSLHFHVRIDVSHFLDONLQGRI
DXWRPRWLYHSDUWSHUPRQWKVXSSOLHUVLQWKHFDULQGXVWU\DUH
DEOHWRVXSSO\WKHVHSDUWV7KHFRPSDQ\DLPVWRVHOHFWDWOHDVW
IRXU RI WKHP DQG WR GHWHUPLQH WKH RSWLPDO DPRXQW RI RUGHUV
DOORFDWHG WR HDFK VXSSOLHU 7KUHH GLIIHUHQW REMHFWLYHV QHHG WR
EHPHWDVIROORZV
0LQLPL]LQJWKHWRWDOSURGXFWLRQWLPH
0LQLPL]LQJWKHWRWDODPRXQWRIVFUDS
0D[LPL]LQJWKHGHOLYHU\UHOLDELOLW\SHUFHQWRIWKHSDUWV
7RWDOYDULDEOHFRVWVKRXOGQRWH[FHHGDERXWIRUHDFK
PRQWK &RQVLGHULQJ WKH LPSUHFLVH GHPDQG SHU PRQWK WKH
WULDQJXODUIX]]\QXPEHU>@LVXVHGLQWKHULJKW
KDQG VLGH RI WKH FRQVWUDLQW UHODWHG WR LW ,Q UHJDUG WR WKH
LPSUHFLVHGHPDQGWKHWRWDOYDULDEOHFRVWZRXOGEHLPSUHFLVH
DQGLVSUHVHQWHGE\WKHWULDQJXODUIX]]\QXPEHU>
@LQWKHFRQVWUDLQWUHODWHGWRYDULDEOHFRVW
'DWD QHHGHG WR EXLOG WKH PRGHO DUH SURYLGHG E\ WKH
FRPSDQ\LQ7DEOH,
,9 352326('0(7+2'72),1'7+()8==<237,0$/
62/87,212)7+(352%/(0
,QWKLVVHFWLRQDQHZPHWKRGLVSURSRVHGWRILQGWKHIX]]\
RSWLPDOVROXWLRQRIWKHW\SHRI)02/33
6WHS&RQVLGHUHDFKREMHFWLYHIXQFWLRQRIWKHPRGHOVXEMHFW
WR DOO FRQVWUDLQWV H[LVW LQ WKH RULJLQDO SUREOHP
VHSDUDWHO\DQGGHILQH3VLQJOHREMHFWLYHSUREOHPV
6WHS8VLQJ DULWKPHWLF RSHUDWLRQV GHILQHG LQ 'HILQLWLRQ WKH VLQJOH REMHFWLYH IX]]\ OLQHDU SURJUDPPLQJ
SUREOHPREWDLQHGLQ6WHSLVFRQYHUWHGLQWR
ஸ
ǡ σ௡௝ୀଵ ܽ௜௝ ‫ݓ‬௝ ൯ ൬ୀஹ൰
7$%/(,
'$7$5(/$7('72($&+&20321<
UDWHRI
YDULDEOHFRVW SURGXFWLRQWLPH
GHOLYHU\UHOLDELOLW\
VFUDSSHU
FRPSDQ\
SHUFHQW
SHUSDUW
SHUSDUW
SDUW
ሺܾ௜௠ ǡ ܾ௜௪ ǡ ܾ௜௪ƴ ሻ‫ ݅׊‬ൌ ݅ ൌ ͳǡ ǥ ǡ ‫ܯ‬
‫ݔ‬෤ ൌ ሺ‫ݔ‬௝௠ ǡ ‫ݓ‬௝ ǡ ‫ݓ‬ƴ௝ ሻ݅‫ ݊݋݊ݏ‬െ ݊݁݃ܽ‫ ݆݁ݒ݅ݐ‬ൌ ͳǡ ǥ ǡ ݊
6WHS8VLQJWKHIX]]\UDQNLQJPHWKRGGHILQHGLQ'HILQLWLRQ
WKHSUREOHPLVFRQYHUWHGWRDFULVS/3SUREOHP
DV
ƒšሺ‹ሻሺ୔୫ ሻ ൌ σ௡௝ୀଵ ܿ௝ ‫ݔ‬௝௠ ‹ሺƒšሻ൫୮୵ ൅ ୮୵ƴ ൯ ൌ σ௡௝ୀଵ ܿ௝ ‫ݓ‬௝ ൅ ܿ௝ ‫ݓ‬ƴ௝ ‫ ܲ׊‬ൌ ͳǡ ǥ ǡ ‫݌‬
ஸ
‫ݏ‬Ǥ ‫ݐ‬Ǥ σ௡௝ୀଵ ܽ௜௝ ‫ݔ‬௝௠ ൬ୀஹ൰ ܾ௜௠ ‫ ݅׊‬ൌ ͳǡ ǥ ǡ ‫ܯ‬
ஹ
σ௡௝ୀଵ ܽ௜௝ ൫‫ݓ‬௝ ൅ ‫ݓ‬ƴ௝ ൯ ൬ୀஸ൰ ܾ௜௪ ൅ ܾ௜௪ƴ ‫ ݅׊‬ൌ ͳǡ ǥ ǡ ‫ܯ‬
‫ݔ‬௝௠ െ ‫ݓ‬௝ ൒ Ͳ݆ ൌ ͳǡ ǥ ǡ 
ஹ
ZKHUH
‫ݏ‬Ǥ ‫ݐ‬Ǥ൫σ௡௝ୀଵ ܽ௜௝ ‫ݔ‬௝௠ ǡ σ௡௝ୀଵ ܽ௜௝ ‫ݓ‬௝
‫ ݅׊‬ൌ ͳǡ ǥ ǡ ‫ܯ‬
σ௡௝ୀଵ ܽ௜௝ ൫‫ݓ‬௝ ൅ ‫ݓ‬ƴ௝ ൯ ൬ୀஸ൰ ܾ݅‫ ݓ‬൅ ܾ௜௪ƴ ‫ ݅׊‬ൌ ͳǡ ǥ ǡ ‫ܯ‬
‫ݔ‬௝௠ െ ‫ݓ‬௝ ൒ Ͳ݆ ൌ ͳǡ ǥ ǡ 
ƒšሺ‹ሻ ܼ෨௉ ൌ ܿ௣ଵ ሺ‫ݔ‬ଵ௠ ǡ ‫ݓ‬ଵ ǡ ‫ݓ‬ƴଵ ሻ ൅ ܿ௣ଶ ሺ‫ݔ‬ଶ௠ ǡ ‫ݓ‬ଶ ǡ ‫ݓ‬ƴଶ ሻ ൅ ‫ ڮ‬൅
ܿ௣௡ ሺ‫ݔ‬௡௠ ǡ ‫ݓ‬௡ ǡ ‫ݓ‬ƴ௡ ሻ‫ ܲ׊‬ൌ ͳǡ ǥ ǡ ‫݌‬
‫ݏ‬Ǥ ‫ݐ‬Ǥ ܽ௜ଵ ሺ‫ݔ‬ெ௠ ǡ ‫ݓ‬ଵ ǡ ‫ݓ‬ƴଵ ሻ ൅ ܽ௜ଶ ሺ‫ݔ‬௠௠ ǡ ‫ݓ‬ଶ ǡ ‫ݓ‬ƴଶ ሻ ൅ ‫ ڮ‬൅
ܽ௜௡ ሺ‫ݔ‬௡௠ ǡ ‫ݓ‬௡ ǡ ‫ݓ‬ƴ௡ ሻ ൑ ሾܾ௜ ǡ ܾ௜௪ ǡ ܾ௜௪ƴ ሿ݅ ൌ ͳǡ ǥ ǡ ‫ܯ‬
ሺ‫ݔ‬௝௠ ǡ ‫ݓ‬௝ ǡ ‫ݓ‬ƴ௝ ሻ ൒ Ͳ݆ ൌ ͳǡ ǥ ǡ ݊
ƒšሺ‹ሻሺ୔୫ ǡ ୔୵ ǡ ୔୵ƴ ሻ ൌ
൫σ௡௝ୀଵ ܿ௝ ‫ݔ‬௝௠ ǡ σ௡௝ୀଵ ܿ௝ ‫ݓ‬௝ ǡ σ௡௝ୀଵ ܿ௝ ‫ݓ‬ƴ௝ ൯ ‫ ܲ׊‬ൌ ͳǡ ǥ ǡ ‫݌‬
‫כ‬
‫כ‬
‫כ‬
‹ σ௉௣ୀଵ ‫ݓ‬௉ ሾሺ୔୫ ǡ ୔୵ ǡ ୔୵ƴ ሻ െ ሺܼ௉௠
ǡ ܼ௉௪
ǡ ܼ௉௪
ƴ ሻሿ
$FFRUGLQJ WR SUREOHP PRGHO LV SURSRVHG WR
VDWLVI\WKHREMHFWLYHVRIWKHFRPSDQ\FRQVLGHULQJLWVGHPDQGV
DQGFRQVWUDLQWV
6WHS)LQG WKH RSWLPDO VROXWLRQ RI DOO 3 VLQJOH REMHFWLYH
SUREOHPV ‫ݔ‬෤௣‫ כ‬XVLQJ WKH &/3 SUHVHQWHG LQ 6WHS DQG
‫כ‬
‫כ‬
‫כ‬
ǡ ܼ௉௪
ǡ ܼ௉௪
WKHQFDOFXODWHܼ௉‫ כ‬ൌ ሺܼ௉௠
ƴ ሻˆ‘”‡ƒ…Š ൌ ͳǡ ǥ ǡ ’
International Scholarly and Scientific Research & Innovation 10(9) 2016
418
scholar.waset.org/1999.7/10005261
World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:10, No:9, 2016
International Science Index, Mathematical and Computational Sciences Vol:10, No:9, 2016 waset.org/Publication/10005261
‹ ܼ෨ଵ ൌ ʹ͵ͻǤͳ͸ሺ‫ݔ‬ଵ௠ ǡ ‫ݓ‬ଵ ǡ ‫ݓ‬ଵ ሻ ൅ ʹͺʹǤͳͶሺ‫ݔ‬ଶ௠ ǡ ‫ݓ‬ଶ ǡ ‫ݓ‬ଶ ሻ ൅
ʹʹͶሺ‫ݔ‬ଷ௠ ǡ ‫ݓ‬ଷ ǡ ‫ݓ‬ଷ ሻ ൅ ʹʹͳǤʹͷሺ‫ݔ‬ସ௠ ǡ ‫ݓ‬ସ ǡ ‫ݓ‬ସ ሻ ൅
ʹ͵ͳǤʹͷሺ‫ݔ‬ହ௠ ǡ ‫ݓ‬ହ ǡ ‫ݓ‬ହ ሻ ൅ ͵͵ͻǤͺሺ‫଺ݔ‬௠ ǡ ‫ ଺ݓ‬ǡ ‫ ଺ݓ‬ሻ ൅
ʹͳ͹Ǥͷሺ‫଻ݔ‬௠ ǡ ‫ ଻ݓ‬ǡ ‫ ଻ݓ‬ሻ ൅ ʹͳͳǤ͸͸ሺ‫଼ݔ‬௠ ǡ ‫ ଼ݓ‬ǡ ‫ ଼ݓ‬ሻ ൅
͵͵͸Ǥʹͷሺ‫ݔ‬ଽ௠ ǡ ‫ݓ‬ଽ ǡ ‫ݓ‬ଽ ሻ ൅ ͵Ͷͺሺ‫ݔ‬ଵ଴௠ ǡ ‫ݓ‬ଵ଴ ǡ ‫ݓ‬ଵ଴ ሻ ‹ ܼ෨ଶ ൌ ͲǤ͵͵͵ሺ‫ݔ‬ଵ௠ ǡ ‫ݓ‬ଵ ǡ ‫ݓ‬ଵ ሻ ൅ ͲǤͳ͸͸ሺ‫ݔ‬ଶ௠ ǡ ‫ݓ‬ଶ ǡ ‫ݓ‬ଶ ሻ ൅
ͲǤͳ͸͸ሺ‫ݔ‬ଷ௠ ǡ ‫ݓ‬ଷ ǡ ‫ݓ‬ଷ ሻ ൅ ͲǤͶͳ͸ሺ‫ݔ‬ସ௠ ǡ ‫ݓ‬ସ ǡ ‫ݓ‬ସ ሻ ൅ ͲǤͷሺ‫ݔ‬ହ௠ ǡ ‫ݓ‬ହ ǡ ‫ݓ‬ହ ሻ ൅
ͲǤ͹ͷሺ‫଺ݔ‬௠ ǡ ‫ ଺ݓ‬ǡ ‫ ଺ݓ‬ሻ ൅ ͲǤͺ͵ሺ‫଻ݔ‬௠ ǡ ‫ ଻ݓ‬ǡ ‫ ଻ݓ‬ሻ ൅ ͲǤͻͷሺ‫଼ݔ‬௠ ǡ ‫ ଼ݓ‬ǡ ‫ ଼ݓ‬ሻ ൅
ͲǤͷሺ‫ݔ‬ଽ௠ ǡ ‫ݓ‬ଽ ǡ ‫ݓ‬ଽ ሻ ൅ ͲǤ͵͵͵ሺ‫ݔ‬ଵ଴௠ ǡ ‫ݓ‬ଵ଴ ǡ ‫ݓ‬ଵ଴ ሻ
ƒš ܼ෨ଷ ൌ ͲǤͻሺ‫ݔ‬ଵ௠ ǡ ‫ݓ‬ଵ ǡ ‫ݓ‬ଵ ሻ ൅ ͲǤ͸ሺ‫ݔ‬ଶ௠ ǡ ‫ݓ‬ଶ ǡ ‫ݓ‬ଶ ሻ ൅
ͲǤͷሺ‫ݔ‬ଷ௠ ǡ ‫ݓ‬ଷ ǡ ‫ݓ‬ଷ ሻ ൅ ͲǤͻሺ‫ݔ‬ସ௠ ǡ ‫ݓ‬ସ ǡ ‫ݓ‬ସ ሻ ൅ ͳǤ͵ሺ‫ݔ‬ହ௠ ǡ ‫ݓ‬ହ ǡ ‫ݓ‬ହ ሻ ൅
ͳǤͳሺ‫଺ݔ‬௠ ǡ ‫ ଺ݓ‬ǡ ‫ ଺ݓ‬ሻ ൅ ͲǤͶሺ‫଻ݔ‬௠ ǡ ‫ ଻ݓ‬ǡ ‫ ଻ݓ‬ሻ ൅ ͲǤ͸ሺ‫଼ݔ‬௠ ǡ ‫ ଼ݓ‬ǡ ‫ ଼ݓ‬ሻ ൅
ͲǤͻሺ‫ݔ‬ଽ௠ ǡ ‫ݓ‬ଽ ǡ ‫ݓ‬ଽ ሻ ൅ ͲǤ͸ሺ‫ݔ‬ଵ଴௠ ǡ ‫ݓ‬ଵ଴ ǡ ‫ݓ‬ଵ଴ ሻ
6XEMHFWWR
͸ǤͲͲͻሺ‫ݔ‬ଵ௠ ǡ ‫ݓ‬ଵ ǡ ‫ݓ‬ଵ ሻ ൅ ͷǤͷ͹ሺ‫ݔ‬ଶ௠ ǡ ‫ݓ‬ଶ ǡ ‫ݓ‬ଶ ሻ ൅
ͶǤͺͻͺሺ‫ݔ‬ଷ௠ ǡ ‫ݓ‬ଷ ǡ ‫ݓ‬ଷ ሻ ൅ ͷǤͳͷʹሺ‫ݔ‬ସ௠ ǡ ‫ݓ‬ସ ǡ ‫ݓ‬ସ ሻ ൅
ͷǤͻͺͷሺ‫ݔ‬ହ௠ ǡ ‫ݓ‬ହ ǡ ‫ݓ‬ହ ሻ ൅ ͸ǤͶͶͶሺ‫଺ݔ‬௠ ǡ ‫ ଺ݓ‬ǡ ‫ ଺ݓ‬ሻ ൅ ͷǤͲͷሺ‫଻ݔ‬௠ ǡ ‫ ଻ݓ‬ǡ ‫ ଻ݓ‬ሻ ൅
ͷǤͺͷ͵ሺ‫଼ݔ‬௠ ǡ ‫ ଼ݓ‬ǡ ‫ ଼ݓ‬ሻ ൅ ͹Ǥʹͺሺ‫ݔ‬ଽ௠ ǡ ‫ݓ‬ଽ ǡ ‫ݓ‬ଽ ሻ ൅
͹Ǥ͵͹ሺ‫ݔ‬ଵ଴௠ ǡ ‫ݓ‬ଵ଴ ǡ ‫ݓ‬ଵ଴ ሻ ൑ ሾͳ͵ͲͲͲͳͲͲͲͳͲͲͲሿ
ሺ‫ݔ‬ଵ௠ ǡ ‫ݓ‬ଵ ǡ ‫ݓ‬ଵ ሻ ൅ ሺ‫ݔ‬ଶ௠ ǡ ‫ݓ‬ଶ ǡ ‫ݓ‬ଶ ሻ ൅ ሺ‫ݔ‬ଷ௠ ǡ ‫ݓ‬ଷ ǡ ‫ݓ‬ଷ ሻ ൅
ሺ‫ݔ‬ସ௠ ǡ ‫ݓ‬ସ ǡ ‫ݓ‬ସ ሻ ൅ ሺ‫ݔ‬ହ௠ ǡ ‫ݓ‬ହ ǡ ‫ݓ‬ହ ሻ ൅ ሺ‫଺ݔ‬௠ ǡ ‫ ଺ݓ‬ǡ ‫ ଺ݓ‬ሻ ൅ ሺ‫଻ݔ‬௠ ǡ ‫ ଻ݓ‬ǡ ‫ ଻ݓ‬ሻ ൅
ሺ‫଼ݔ‬௠ ǡ ‫ ଼ݓ‬ǡ ‫ ଼ݓ‬ሻ ൅ ሺ‫ݔ‬ଽ௠ ǡ ‫ݓ‬ଽ ǡ ‫ݓ‬ଽ ሻ ൅ ሺ‫ݔ‬ଵ଴௠ ǡ ‫ݓ‬ଵ଴ ǡ ‫ݓ‬ଵ଴ ሻ ൑
ሾʹͶͲͲʹͲͲʹͲͲሿ ‫ݔ‬௝௠ െ ‫ݓ‬௝ ൒ ͳͲͲ‫ݕ‬௝ ‫ݔ‬௝௠ െ ‫ݓ‬௝ ൑ ʹ͸ͲͲ‫ݕ‬௝ ݆ ൌ ͳǡ ǥ ǡͳͲ
‫ݕ‬ଵ ൅ ‫ݕ‬ଶ ൅ ‫ݕ‬ଷ ൅ ‫ݕ‬ସ ൅ ‫ݕ‬ହ ൅ ‫ ଺ݕ‬൅ ‫ ଻ݕ‬൅ ‫ ଼ݕ‬൅ ‫ݕ‬ଽ ൅ ‫ݕ‬ଵ଴ ൒ ͷ
‫ݔ‬௝௠ ൒ Ͳܽ݊݀݅݊‫ݓݎ݁݃݁ݐ‬௝ ൒ Ͳܽ݊݀݅݊‫ ݆ݎ݁݃݁ݐ‬ൌ ͳǡ ǥ ǡͳͲ
‫ݕ‬ଵ ǡ ‫ݕ‬ଶ ǡ ‫ݕ‬ଷ ǡ ‫ݕ‬ସ ǡ ‫ݕ‬ହ ǡ ‫ ଺ݕ‬ǡ ‫ ଻ݕ‬ǡ ‫ ଼ݕ‬ǡ ‫ݕ‬ଽ ǡ ‫ݕ‬ଵ଴ ൌ Ͳ‫ͳݎ݋‬
ZKHUH‫ݔ‬෤௝ ൌ ݂‫ݎ݈݁݅݌݌ݑݏ݄ݐ݆݋ݐ݀݁ݐܽܿ݋݈݈ܽݏݎ݁݀ݎ݋݂݋ݎܾ݁݉ݑ݊ݕݖݖݑ‬
7KH ILQDO UHVXOW REWDLQHG IURP VROYLQJ WKH PRGHO LV
SUHVHQWHGEHORZ
š෤ଵ ൌ ሺͳͲͲǡͲǡͲሻš෤ଶ ൌ ሺͳͲͲǡͲǡͲሻš෤ସ ൌ ሺͳͷͺ͹ǡͲǡͲሻš෤ହ ൌ
ሺ͸ͳ͵ǡʹͲͲǡʹͲͲሻ
෨ ‫כ‬ଵ ൌ ሺͷͳ͹Ͳ͹ͺǤͳͶǡͶ͸ʹͷͲǡͶ͸ʹͷͲሻ
෨ ‫ כ‬ଶ ൌ ሺͳͲͳ͸ǤͷͻʹǡͳͲͲǡͳͲͲሻ
෨ ‫ כ‬ଷ ൌ ሺʹ͵͹ͷǤʹǡʹ͸Ͳǡʹ͸Ͳሻ
5HVXOWVVKRZWKDWVXSSOLHUVDQGDUHWKHEHVWRSWLRQV
IRUWKHFRPSDQ\DQGWKHDPRXQWRIRUGHUVDOORFDWHGWRWKHP
DQG WKH DFFHSWDEOH WROHUDQFH RI GHFLVLRQV DFFRUGLQJ WR WKH
IX]]LQHVV LQ WKH FRPSDQ\ DUH GHWHUPLQHG 6R WKH
LPSUHFLVHQHVVRIWKHGHFLVLRQVLVSUHVHQWHGSUHFLVHO\DQGWKHUH
LVQRQHHGWRGRVHQVLWLYLW\DQDO\VLVRIWKHLQH[DFWULJKWKDQG
VLGH QXPEHUV 7KH RZQHUV RI WKH FRPSDQ\ FRQILUPHG WKH
UHVXOWVDVUHDVRQDEOH
9, &21&/86,21
,QWKLVSDSHUDQHZPHWKRGZDVSURSRVHGWRILQGWKHIX]]\
RSWLPDO VROXWLRQ RI )02/33 ZLWK IX]]\ ULJKW KDQG VLGH DQG
IX]]\ GHFLVLRQ YDULDEOHV %\ XVLQJ WKH SURSRVHG PHWKRG WKH
IX]]\ RSWLPDO VROXWLRQ RI )02/33 ZKLFK RFFXUV LQ UHDO OLIH
VLWXDWLRQFDQEHHDVLO\REWDLQHG:LWKWKHDLGRIWKLVPHWKRG
GHFLVLRQ PDNHUV FDQ REWDLQ IX]]\ GHFLVLRQV WR UHIOHFW WKH
LQKHUHQW IX]]LQHVV RI D GHFLVLRQ SUREOHP DQG WKH QHHG WR GR
VHQVLWLYLW\ DQDO\VLV DIWHU REWDLQLQJ D FULVS VROXWLRQ GHFUHDVHV
7RLOOXVWUDWHWKHSURSRVHGPHWKRGDUHDOOLIHRUGHUDOORFDWLRQ
SUREOHPWRVRPHVXSSOLHUVZDVVROYHGDQGWKHUHVXOWVVDWLVILHG
WKHRZQHUVRIWKHFRPSDQ\
‫ݕ‬௝ ൌ Ͳ‫ͳݎ݋‬ǡ ‫ܼݐ݋݊ݎ݋ݎ݈݁݅݌݌ݑݏ݄ݐ݆݂݋݊݋݅ݐ݈ܿ݁݁ݏ݂݋݈ܾ݁ܽ݅ݎܽݒݕݎܾ݄ܽ݊݅݁ݐ‬෨ଵ WKHWRWDOSURGXFWLRQWLPHܼ෨ଶ WRWDODPRXQWRIVFUDSܼ෨ଷ WRWDOGHOLYHU\UHOLDELOLW\SHUFHQWRIDOOSDUWV
&RQVWUDLQWRWDOGHPDQGRISDUWV
&RQVWUDLQWRWDOYDULDEOHFRVW
&RQVWUDLQLIMWKVXSSOLHULVVHOHFWHGWKHQXPEHURILWVRUGHU
DOORFDWLRQZLOOEHQHLWKHUOHVVWKDQQRUPRUH
WKDQ
$FFRUGLQJWRWKHFULVS02/3LVUHSUHVHQWHGLQ
5()(5(1&(6
>@ *X=KDQJ<+:X05HPLDV-/X)RUPXODWLRQRIIX]]\OLQHDU
SURJUDPPLQJ SUREOHPV DV IRXUREMHFWLYH FRQVWUDLQHG RSWLPL]DWLRQ
SUREOHPV$SSOLHG0DWKHPDWLFVDQG&RPSXWDWLRQ±
>@ ) $UÕNDQ = ODO *XQJRU $ WZRSKDVH DSSURDFK IRU PXOWLREMHFWLYH
SURJUDPPLQJ SUREOHPV ZLWK IX]]\ FRHIILFLHQWV ,QIRUPDWLRQ 6FLHQFHV
±
>@ /$=DGHK)X]]\VHWV,QIRUPDWLRQDQG&RQWURO±
>@ +7DQDND72NXGD.$VDL2QIX]]\PDWKHPDWLFDOSURJUDPPLQJ-
&\EHUQHWLFV6\VW±
>@ +- =LPPHUPDQQ )X]]\ SURJUDPPLQJ DQG OLQHDU SURJUDPPLQJ ZLWK
VHYHUDOREMHFWLYHIXQFWLRQV)X]]\6HWVDQG6\VWHPV±6
>@ - / 9HUGHJD\ )X]]\ PDWKHPDWLFDO SURJUDPPLQJ LQ %0
>@ % :DUQHUV ,QWHUDFWLYH 0XOWLSOH 2EMHFWLYH 3URJUDPPLQJ VXEMHFW WR
IOH[LEOHFRQVWUDLQW(XURSHDQ-RXUQDORI2SHUDWLRQDO5HVHDUFK
>@ %:DUQHUV$Q,QWHUDFWLYH)X]]\3URJUDPPLQJV\VWHP)X]]\6HWVDQG
6\VWHPV
>@ + - =LPPHUPDQ 'HVFULELQJ DQ RSWLPL]DWLRQ RI IX]]\ V\VWHPV
LQWHUQDWLRQDO-RXUQDORI*HQHUDO6\VWHP
>@ 6 &KDQDV WH XVH RI SDUDPHWULF SURJUDPPLQJ LQ )/3 )X]]\ VHWV DQG
6\VWHPV
>@ -/9HUGHJD\$GXDODSSURDFKWRVROYHWKHIX]]\/LQHDU3URJUDPPLQJ
3UREOHP)X]]\6HWVDQG6\VWHPV
>@ - / 9HUGHJD\ $SSOLFDWLRQ RI IX]]\ RSWLPL]DWLRQ LQ RSHUDWLRQDO
UHVHDUFK&RQWURODQG6\EHUQHWLFV
>@ & &DUOVVRQ DQG 3 &RUKRQHQ $ SDUDPHWULF DSSURDFK WR IX]]\ OLQHDU
SURJUDPPLQJ)X]]\6HWVDQG6\VWHPV
>@ - 5DPLN DQG - 5LPDQHN ,QHTXDOLW\ UHODWLRQ EHWZHHQ IX]]\ QXPEHUV
DQGLWVXVHLQIX]]\RWLPL]DWLRQIX]]\6HWVDQG6\VWHPV
ƒš ൌ ͳ͵ͷǤ͸͸ šଵ୫ ൅ ͳͺͷǤͳͶšଶ୫ ൅ ͳͷ͹šଷ୫ ൅ ͳͷͻǤʹͷšସ୫ ൅
ͻͳǤʹͷšହ୫ ൅ ͵ͺͶǤͺš଺୫ ൅ ͷͳʹǤʹͷš଻୫ ൅ ͷͲ͸Ǥ͸͸š଼୫ ൅
͵ͳ͸Ǥʹͷšଽ୫ ൅ ͵͵ͶǤͷšଵ଴୫ ‹ ൌ ͸͹ͷǤ͸͸™ଵ ൅ ͷͶͷǤͳͶ™ଶ ൅ Ͷͷ͹™ଷ ൅ ͸ͻͻǤʹͷ™ସ ൅
ͺ͹ͳǤʹͷ™ହ ൅ ͵ͺͶǤͺ™଺ ൅ ͹ͷʹǤͷ™଻ ൅ ͺ͸͸Ǥ͸͸™଼ ൅ ͺͷ͸Ǥʹͷ™ଽ ൅
͸ͻͶǤͷ™ଵ଴ VW
͸ǤͲͲͻšଵ୫ ൅ ͷǤͷ͹šଶ୫ ൅ ͶǤͺͻͺšଷ୫ ൅ ͷǤͳͷʹšସ୫ ൅ ͷǤͻͺͷšହ୫ ൅
͸ǤͶͶͶš଺୫ ൅ ͷǤͲͷš଻୫ ൅ ͷǤͺͷ͵š଼୫ ൅ ͹Ǥʹͺšଽ୫ ൅ ͹Ǥ͵͹šଵ଴୫ ൑
ͳ͵ͲͲͲ
͸ǤͲͲͻ™ଵ ൅ ͷǤͷ͹™ଶ ൅ ͶǤͺͻͺ™ଷ ൅ ͷǤͳͷʹ™ସ ൅ ͷǤͻͺͷ™ହ ൅
͸ǤͶͶͶ™଺ ൅ ͷǤͲͷ™଻ ൅ ͷǤͺͷ͵™଼ ൅ ͹Ǥʹͺ™ଽ ൅ ͹Ǥ͵͹™ଵ଴ ൒ ͳͲͲͲ šଵ୫ ൅ šଶ୫ ൅ šଷ୫ ൅ šସ୫ ൅ šହ୫ ൅ š଺୫ ൅ š଻୫ ൅ š଼୫ ൅ šଽ୫ ൅
šଵ଴୫ ൑ ʹͶͲͲ
™ଵ ൅ ™ଶ ൅ ™ଷ ൅ ™ସ ൅ ™ହ ൅ ™଺ ൅ ™଻ ൅ ™଼ ൅ ™ଽ ൅ ™ଵ଴ ൒ ʹͲͲ
š୨୫ െ ™୨ ൒ ͳͲͲ›୨ Œ ൌ ͳǡʹǡ ǥ ǡͳͲ
š୨୫ െ ™୨ ൑ ʹ͸ͲͲ›୨ Œ ൌ ͳǡʹǡ ǥ ǡͳͲ
›ଵ ൅ ›ଶ ൅ ›ଷ ൅ ›ସ ൅ ›ହ ൅ ›଺ ൅ ›଻ ൅ ›଼ ൅ ›ଽ ൅ ›ଵ଴ ൒ ͷ
š୨୫ ൒ Ͳƒ†‹–‡‰‡”™୨ ൒ Ͳƒ†‹–‡‰‡”Œ ൌ ͳǡ ǥ ǡͳͲ
International Scholarly and Scientific Research & Innovation 10(9) 2016
419
scholar.waset.org/1999.7/10005261
World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:10, No:9, 2016
International Science Index, Mathematical and Computational Sciences Vol:10, No:9, 2016 waset.org/Publication/10005261
>@ + 7DQDND + ,FKLKDVKL DQG . $VDL $ IRUPXODWLRQ RI IX]]\ /LQHDU
3URJUDPPLQJ3UREOHPVEDVHGRQFRPSDULVRQRIIX]]\QXPEHUV&RQWURO
DQGF\EHUQHWLFV
>@ <-/DLDQG&/+ZDQJ$QHZDSSURDFKWRVRPHSRVVLELOLVWLFOLQHDU
SURJUDPPLQJSUREOHPV)X]]\6HWVDQG6\VWHPV
>@ + 5RPPHOIDQJHU 5 +DQXVFKHFN DQG - :ROI /LQHDU SURJUDPPLQJ
ZLWKIX]]\REMHFWLYHV)X]]\6HWVDQG6\VWHPV
>@ -- %XFNOH\ 3RVVLELOLVWLF /LQHDU SURJUDPPLQJ ZLWK WULDQJXODU IX]]\
QXPEHUV)X]]\6HWVDQG6\VWHPV
>@ & 6WDQFLXOHVFX 3K )RUWHPSV 0 ,QVWDOO 9 :HUW] 0XOWLREMHFWLYH
IX]]\ OLQHDU SURJUDPPLQJ SUREOHPV ZLWK IX]]\ GHFLVLRQ YDULDEOHV
(XURSHDQ-RXUQDORI2SHUDWLRQDO5HVHDUFK±
>@ +7DQDND3*XR+-=LPPHUPDQQ3RVVLELOLW\GLVWULEXWLRQVRIIX]]\
GHFLVLRQ YDULDEOHV REWDLQHG IURP SRVVLELOLVWLF OLQHDU SURJUDPPLQJ
SUREOHPV)X]]\6HWVDQG6\VWHPV
>@ +5 0DOHNL 0 7DWD 0 0DVKLQFKL /LQHDU SURJUDPPLQJ ZLWK IX]]\
YDULDEOHV)X]]\6HW6\VW±
>@ - %XFNOH\ 7 )HXULQJ (YROXWLRQDU\ DOJRULWKP VROXWLRQ WR IX]]\
SUREOHPV IX]]\ OLQHDU SURJUDPPLQJ )X]]\ 6HW 6\VW ±
>@ 60 +DVKHPL 0 0RGDUUHV ( 1DVUDEDGL 00 1DVUDEDGL )XOO\
IX]]LILHGOLQHDUSURJUDPPLQJVROXWLRQDQGGXDOLW\-,QWHOO)X]]\6\VW
±
>@ 7 $OODKYLUDQORR )+ /RWIL 0. .LDVDU\ 1$ .LDQL / $OL]DGHK
6ROYLQJIXOOIX]]\OLQHDUSURJUDPPLQJSUREOHPE\WKHUDQNLQJIXQFWLRQ
$SSO0DWK6FL±
>@ $.XPDU-.DXU36LQJK$QHZPHWKRGIRUVROYLQJIXOO\IX]]\OLQHDU
SURJUDPPLQJ SUREOHPV $SSOLHG 0DWKHPDWLFDO 0RGHOOLQJ ±
>@ 0 'HKJKDQ % +DVKHPL 0 *KDWHH &RPSXWDWLRQDO PHWKRGV IRU
VROYLQJ IXOO\ IX]]\ OLQHDU V\VWHPV $SSO 0DWK &RPSXW ±
>@ )+ /RWIL 7 $OODKYLUDQORR 0$ -RQGDEHKD / $OL]DGHK 6ROYLQJ D
IXOO\ IX]]\ OLQHDU SURJUDPPLQJ XVLQJ OH[LFRJUDSK\ PHWKRG DQG IX]]\
DSSUR[LPDWHVROXWLRQ$SSO0DWK0RGHOO±
International Scholarly and Scientific Research & Innovation 10(9) 2016
420
scholar.waset.org/1999.7/10005261