Post on 04-Feb-2021
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karRsav RCav Rbt ibt þikar2
Operations Research II kRmitmUldæan
y wm Gay uv DÆn³v iC¢a
qñaM 20 13
karRsav RCav Rbt ibt þikar2
Operations Research II
kRmitmUldæan
i
GarmÖkfa
esovePA “karRsavRCavRbtibtþikar2 ” enHeroberogeLIg edIm,ITukCaÉksarCMnYyxøH²
dl;karRsavRCavrbs;saRsþacarü nisSit nigGñkRsavRCavmYycMnYnEdlmanbMNgsikSamux
CMnajKNitviTüaGnuvtþn_.
naeBlbc©úb,nñenH GñkRKb;RKg b¤GñkdwknaMTaMgLay CaBiessGñkRKb;RKgxagEpñk
plitkmµEtgEtriHrkmeFüa)ayeFVIy:agNaedIm,I[karRKb;RKgrbs;BYkeKRbkbedayRbsiT§PaB
EdlQaneTArkkarTTYl)annUvPaBRbesIrbMputénR)ak;cMeNj b¤éføedImkñúgplitkmµ b¤esva
kmµenAkñúgGgÁPaBrbs;BYkeK dUcenHedIm,I[karRKb;RKgmanRbsiT§PaB enaHeKRtUveRbIviFIviTüa
sa®sþTaMgLayenAkñúgmuxviC¢a “karRsavRCavRbtibtþikar2 ” enH ehIymü :ageToteyIgBuMTan;
manÉksarRKb;RKan;kñúgkarRsavRCav CaBiessÉksarCaexmrPasa. eyageTAelItRmUvkarenH
ehIyEdlmuxviC¢aenHRtUv)anelIkykmksikSaedIm,ICaCMnYykñúgkareFVIseRmccitþ.
esovePAenHRtUv)aneroberogeLIgedayEckecjCa5CMBUk edayCMBUkTI1nwgBnül;GMBI
karbEnßmplitplfµI CMBUkTI2nwgsikSaGMBIcMeNaTeTVPaB CMBUkTI3nwgsikSaGMBIkarviPaKeRkay
cemøIyRbesIrbMput CMBUkTI4nwgsikSaGMBIcMeNaTkardwkCBa¢Ún nigCMBUkTI5nwgsikSaGMBIRkahVman
TisedA nigbNþaj.
GñkeroberogsgÇwmfa muxviC¢aenHnwgpþl;nUvcMeNHdwgCaeRcIndl;saRsþacarü nisSit Gñk
RKb;RKgnigGñkRsavRCavTaMgLay ehIysUmsVaKmn_Canic©ral;karriHKn;sßabna edIm,I[esovePA
enHkan;EtsuRkitüEfmeTot.
ii
mat ika
TMB½r
GarmÖkfa .......................................................................................................... i
mat ika ............................................................................................................. ii
CMBUkT I1 karbEnßmplitplfµI .................................................................. 1
I- kareRbIR)as;ka)a:sIuFIbRmug ........................................................................ 1
II- karsg;taragfµI ............................................................................................. 2
lMhat ; ..................................................................................................... 6
CMBUkT I2 cMeNaTeTVPaB ........................................................................... 9
I- cMeNaTdMbUg nigcMeNaTeTVPaB ............................................................... 9
II- cemøIyRbesIrbMputéncMeNaTeTVPaB ...................................................... 12
lMhat ; ................................................................................................... 16
CMBUkT I3 karv iPaKeRkay cemøIy RbesIrbMput ......................................... 19
I- esckþIepþIm .................................................................................................... 19
II- karv iPaKR)ak;cMeNjÉkta ...................................................................... 20
lMhat ; ................................................................................................... 21
CMBUkT I4 cMeNaTkardwkCBa¢Ún .................................................................. 24
I- esckþIepþIm ................................................................................................. 24
II- cemøIydMbUgéncMeNaTkardwkCBa¢Ún .......................................................... 28
III- v iFIrktémøRbEhlv :UEhÁl ....................................................................... 31
iii
IV- v iFIsþibPIgsþÚn³ rkcemøIyéføed ImTabbMput ........................................... 32
V- »nPaB ...................................................................................................... 37
VI- cemøIyRbesIrbMputmaneRcIn nigbERmbRmÜléføed ImÉkta .................. 38
lMhat ; ................................................................................................... 39
CMBUkT I5 RkahVman T isedA n igbNþaj ............................................ 42
I- esckþIepþIm .................................................................................................... 42
II- karR)aRs½yTak;Tg ................................................................................... 43
III- karRbkYtRbECg ....................................................................................... 43
IV- cMeNaTGb,brmaénbNþaj ................................................................... 44
V- cMeNaTrkRbEvgpøÚv xøIbMput ....................................................................... 45
VI- cMeNaTrklMhUrGtibrma ........................................................................ 45
lMhat ; ................................................................................................... 46
ÉksarBieRKaH ............................................................................................. 48
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
1
CMBUkT I1 karbEn ßmplitplfµI (Adding New Product)
I- kareRbIR)as;ka)a:sIuFIbRmug ( Using Available Capacities )
sar³sMxan;enAkñúgCMBUkenH KWeFVIkarseRmccitþfa etIKYrplitplitplfµIbEnßmeTotb¤
y:agNa. eyIgBinitüemIlcMeNaTkmµviFIlIenEG‘rmYyeLIgvij.
]TahrN_TI1 (kñúgbBaðaplitplcRmuH)
shRKaskat;edrmYy mansgVak;plitkmµBIrlMdab;KW kat; nigedr . kñúgmYyéf¶² eKman
eBlevlabRmugsRmab;kat; 150 h nigeBlevlabRmugedr 200 h . plitplrbs;shRKasKWCa
exa nigGavEdleKGacrk)anR)ak;cMeNj $5 kñúgexamYy nig $4 kñúgGavmYy. kargarkat;kñúgmYy
ema:g²eKGackat;)anexa 8 b¤Gav 6 . kic©karedrRtUvcMNayeBl 15 mn sRmab;examYy nig
10 mn sRmab;GavmYy. etIkñúgmYyéf¶shRKasRtUvplitexa nigGavb:unµan edIm,I)anR)ak;cMeNj
Gtibrma?
tag x1 CacMnYnexaEdlRtUvplit (x1 ≥ 0) nigtag x2 CacMnYnGavEdlRtUvplit (x2 ≥ 0).
eK)anragsþg;da b¤bRgYm ( The standard Form ) :
Maximize profit z = 5 x1 + 4 x2
Subject to (1/8)x1 + (1/6)x2 ≤ 150 ( 1 ) : Cutting time constraint
(1/4)x1 + (1/6)x2 ≤ 200 ( 2 ) : Sewing time constraint
x1, x2 ≥ 0 : Non negativity constraint
edaHRsaycMeNaTenHtamviFIsIumepøc eK)antaragsIumepøccugeRkaydUcxageRkam³
Table 1.1 Basic x1 x2 S1 S2 z RHS
x2
x1
0
1
1
0
12 – 6
8 – 8
0
0
600
400
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
2
OBJ 0 0 8 1 16 4400
cemøIyRbesIrbMput (Optimal Solution ) KW ³
x1 = 400, x2 = 600, S1 = S2 = 0 nig Max z = 4400 .
]bmafa Epñk Marketing pþl;B½t’manfa ebIeKplitsMBt; eKGacrk)anR)ak;cMeNj $5 kñúg
sMBt;mYy. karplitsMBt;RtUvcMNayeBlevlakat; 8 mn nigeBlevlaedr 5 mn kñúgmYyÉkta.
etIeKRtUvplitsMBt;b¤eT?
taragcugeRkaybgðaj[dwgfa
S1 = 0 : eBlevlabRmugsRmab;kat; 150 h eRbIR)as;Gs;ehIy nig
S2 = 0 : eBlevlabRmugsRmab;edr 200 h k¾eRbIR)as;Gs;ehIyEdr.
eKrMBwgfa karplitsMBt;R)akdCaeFVI[cMnYnexa nigGavfycuH.
l½kçx½NÐénkarseRmccitþ³ etIKYrEtplitsMBt;b¤eT? eKBinitüemIlplb:HBal;elIR)ak;
cMeNj enAeBleKplitsMBt;mYyÉkta.
- plitsMBt;mYycMNayeBlkat; 8 mn = 2/15 h enaH S1 fyGs; 2/15 Ékta ehIyeK
)an z fyGs; 2/15 × 8 = 16/15 = $1.07 .
- plitsMBt;mYycMNayeBledr 5 mn = 1/12 h enaH S2 fyGs; 1/12 Ékta ehIyeK
)an z fyGs; 1/12 × 16 = 4/3 = $1.33 .
eyIgeXIjfa karplitsMBt;mYyeFVI[R)ak;cMeNjsrubfyGs; $1.07 + $1.33 = $2.4
b:uEnþsMBt;mYyenHGacpþl;R)ak;cMeNjmkvijdl;eTA $5 enaH z ekIn)an $5 – $2.4 = $2.6 .
karseRmccitþ³ eKKb,IplitsMBt;mYymuxbEnßmeTot.
II- karsg;taragfµI ( Constructing New Table )
bBaðaecaTcMeBaHshRKasenAeBlenH KWRtUvplitexa Gav nigsMBt;cMnYnb:unµanedIm,I
[)anR)ak;cMeNjGtibrma.
tag x3 CacMnYnsMBt;EdlRtUvplit ( x3 ≥ 0 ) . eK)anragsþg;dafµI
Maximize profit z = 5 x1 + 4 x2 + 5 x3 Subject to (1/8)x1 + (1/6)x2 + (2/15)x3 ≤ 150 ( 1 ) : Cutting time constraint
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
3
(1/4)x1 + (1/6)x2 + (1/12)x3 ≤ 200 ( 2 ) : Sewing time constraint and x1, x2, x3 ≥ 0 : Non negativity constraint
edayBMucaM)ac;edaHRsaytamviFIsIumepøceLIgvij eyIgRKan;EtbegáItCYrQrfµI x3 kñúgtarag
suImepøccugeRkayEdlmanRsab;. edIm,IbMeBjtémøelxkñúgCYrQrfµIenH eKeFVIvicardUcxageRkam³
plit x3 mYyÉktanaM[
S1 fy 2/15 Ékta enaHeK)an x2 fy (2/15) × 12 nig x1 ekIn (2/15) × 8 .
S2 fy 1/12 Ékta enaHeK)an x2 ekIn (1/12) × 6 nig x1 fy (1/12) × 8 .
- plb:HBal;elI x2 : – [(2/15) × 12] + (1/12) × 6 = – 11/10 mann½yfa x2 fyGs;
11/10 Ékta rYcsresrcMnYn 11/10 kñúgCYredk x2 énCYrQrfµI x3 .
- plb:HBal;elI x1 : (2/15) × 8 – [(1/12) × 8] = 2/5 mann½yfa x1 ekIn)an 2/5 Ékta
rYcsresrcMnYn – 2/5 kñúgCYredk x1 énCYrQrfµI x3 .
- plb:HBal;elIR)ak;cMeNj z : eyIg)anviPaKxagelIrYcehIyfa z ekIn)an
5 – (16/15 + 4/3) = 13/5 = 2.6 rYcsresrcMnYn – 13/5 kñúgCYredk OBJ énCYrQr x3 .
eK)antaragsIumepøcfµImYymanCYrQr x3 EdlminEmnCataragcugeRkayeT BIeRBaHenACYr
edk OBJ mancMnYnGviC¢man – 13/5 .
Table 1.2 Basic x1 x2 x3 S1 S2 z RHS
x2
x1
0
1
1
0
11/10
– 2/5
12
– 8
– 6
8
0
0
600
400
OBJ 0 0 – 13/5 8 16 1 4400
eRbIviFIsIumepøcedIm,IbEmøgtaragfµIenH[eTACataragsuImepøccugeRkay³
Table 1.3 Basic x1 x2 X3 S1 S2 z RHS
x3
x1
0
1
10/11
4/11
1
0
120/11
40/11
– 60/11
64/11
0
0
545.45
618.18
OBJ 0 24/11 0 400/11 20/11 1 5818.17
BItaragcugeRkayenH eKTaj)ancemøIyGubTIm:al;éncMeNaTKW³
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
4
x1 = 618.18, x2 = 0, x3 = 545.45, S1 = 0, S2 = 0 nig Max z = 5818.17
Basic
.
]TahrN_TI2³ kñúgbBaðaplitplcRmuHmYyRtUv)aneKedaHRsaytamviFIsuImepøcCabnþbnÞab; ehIyeK
TTYl)antaragsIumepøccugeRkay³
Table 1.4 x1 x2 S1 S2 S3 z RHS
S1
x2
x1
0
0
1
0
1
0
1
0
0
– 0.4
2
– 2
0.1
– 2
3
0
0
0
60
300
300
OBJ 0 0 0 2 1 1 2100
enAkñúgenH plitpl A manbrimaN x1 pþl;R)ak;cMeNjkñúgmYyÉkta $3 nigplitpl B
manbrimaN x2 pþl;R)ak;cMeNjkñúgmYyÉkta $4 .
eKBinitünUvlT§PaBplitplitplfµI C mYymuxeTotEdlmanR)ak;cMeNjkñúgmYyÉkta $5
ehIymYyÉktaén C RtUveRbIR)as;ka)a:suIFI (1) cMnYn 2 Ékta ka)a:suIFI (2) cMnYn 1.5 Ékta nig
ka)a:suIFI (3) cMnYn 0.5 .
lT§PaBplitplitplfµI³ plitpl C mYyÉkta naM[R)ak;cMeNjedImfycuHcMnYn (2 ×0)
+ (1.5 × 2) + (0.5 × 1) = $3.5 . b:uEnþplitpl C mYyÉktaGacpþl;R)ak;cMeNjmkvij $5 enaH
z ekIn)an $5 – $3.5 = $1.5 . dUcenH eKRtUvplitplitpl C .
sMNg;taragfµI³ tag x3 CabrimaNplitpl C EdlRtUvplit. eyIgbegáItCYrQrfµI x3
bEnßmkñúgtarag. plitpl C mYyÉktaeRbIR)as; S1 = 2 Ékata S2 = 1.5 Ékta nig S3 = 0.5
Ékta. eK)anplb:HBal;elIGefreKaldUcxageRkam³
- plb:HBal;elI S1 : – (2 × 1) + (1.5 × 0.4) – (0.5 × 0.1) = – 1.45
- plb:HBal;elI x2 : (2 × 0) – (1.5 × 2) – (0.5 × 2) = – 2
- plb:HBal;elI x1 : (2 × 0) + (1.5 × 2) – (0.5 × 3) = 1.5
- plb:HBal;elI z : z ekIn)an $1.5 .
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
5
eKTTYl)antaragfµIxageRkam³
Table 1.5 Basic x1 x2 x3 S1 S2 S3 z RHS
S1
x2
x1
0
0
1
0
1
0
1.45
2
– 1.5
1
0
0
– 0.4
2
– 2
0.1
– 2
3
0
0
0
60
300
300
OBJ 0 0 – 1.5 0 2 1 1 2100
eRbIviFIsIumepøcedIm,IbEmøgtaragfµIenH[eTACataragsuImepøccugeRkay³
Table 1.6 Basic x1 x2 x3 S1 S2 S3 z RHS
x3
x2
x1
0
0
1
0
1
0
1
0
0
0.69
– 1.38
1.035
– 0.27
2.54
– 1.595
0.068
– 2.136
3.102
0
0
0
41.38
217.24
362.07
OBJ 0 0 0 1.035 4.025 2.19 1 2162.07
BItaragcugeRkayenH eKTaj)ancemøIyGubTIm:al;éncMeNaTKW³
x1 = 362.07, x2 = 217.24, x3 = 41.38, S1 = 0, S2 = 0, S3 = 0 nig Max z = 2162.07.
♣ ♣♣ ♣♣
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
6
lMhat ;
1. cMeNaTkmµviFIlIenEG‘rmYymanGnuKmn_eKalbMNg z = 3 x + 2 y EdlRtUvrktémøGtibrma.
eK)anedaHRsaytamviFIsuImepøc ehIyTTYl)antaragsIumepøccugeRkay³
Basic x y S1 S2 z RHS
x
S2
1
0
1
3
1
– 1
0
1
0
0
10
2
OBJ 0 1 3 0 1 30
eKBinitünUvlT§PaBplitplitplfµImYymuxeTotEdlGacrk)ancMeNjkñúgmYyÉkta $4 .
karplitplitplfµImYyÉktaRtUveRbIR)as;ka)a:sIuFI (1) cMnYn 2 Ékta nigka)a:sIuFI (2) cMnYn 3
Ékta .
k. etIeKKYrplitplitplfµIenHb¤eT?
x. etIR)ak;cMeNjmYyÉktamankRmitb:unµaneTIbeKGacbþÚrkarseRmccitþ?
K. ]bmaR)ak;cMeNjmYyÉktaénplitplfµIKW $6.5 . begáIttaragEdlmanCYrQrbEnßm
sRmab;plitplfµIenH nigrkcemøIyRbesIrbMput.
2. cMeNaTkmµviFIlIenEG‘rmYymanGnuKmn_eKalbMNg z = 32x1 + 18x2 + 19x3 EdlRtUvrktémø
Gtibrma. eK)anedaHRsaytamviFIsuImepøc ehIyTTYl)antaragsIumepøccugeRkayxageRkam³
Basic x1 x2 x3 S1 S2 S3 z RHS
x1
S2
x3
1
0
0
–1/2
8
2
0
0
1
3/2
– 6
– 2
0
1
0
–1/2
1
1
0
0
0
3
48
6
OBJ 0 4 0 10 0 3 1 210
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
7
eKBinitünUvlT§PaBedIm,IplitplitplfµImYymuxeTotEdlGacrk)anR)ak;cMeNj $25 kñúg
mYyÉkta. karplitplitplfµImYyÉktaRtUveRbIR)as;ka)a:sIuFI (1) cMnYn 2 Ékta ka)a:sIuFI (2)
cMnYn 5 Ékta nigka)a:sIuFI (3) cMnYn 3 Ékta .
k. etIeKKYrplitplitplfµIenHb¤eT?
x. etIR)ak;cMeNjmYyÉktaénplitplfµIenHmankRmitb:unµaneTIbeKGacbþÚrkarseRmccitþ?
K. ]bmaR)ak;cMeNjmYyÉktaénplitplfµIKW $30 . etIeKKYrplitplitplfµIenHb¤eT?
begáIttaragsuImepøcfµI nigrkcemøIyRbesIrbMput.
3. cMeNaTkmµviFIlIenEG‘rmYymanGnuKmn_eKalbMNg z = 6x1 + 10x2 + 8x3 + 3x4 EdlRtUvrk
témøGtibrma. eK)anedaHRsaytamviFIsuImepøc ehIyTTYl)antaragsIumepøccugeRkay³
Basic x1 x2 x3 x4 S1 S2 S3 S4 z RHS
S1
S2
S3
x2
– 1
– 10
3
2
0
0
0
1
– 2
– 10
4
3
3
– 21
– 3
4
1
0
0
0
0
1
0
0
0
0
1
0
– 1
– 6
– 1
1
0
0
0
0
46
23
33
2
OBJ 14 0 22 37 0 0 0 10 1 20
eKBinitünUvlT§PaBplitplitplfµImYymuxeTotEdlGacrk)ancMeNjkñúgmYyÉkta$15.
karplitplitplfµImYyÉktaRtUveRbIR)as;ka)a:sIuFI (1) cMnYn 2 Ékta ka)a:sIuFI (2) cMnYn 1 Ékta
ka)a:sIuFI (3) cMnYn 5 Ékta nigka)a:sIuFI (4) cMnYn 2 Ékta.
k.etIeKKYrplitplitplfµIenHEdrb¤eT? begáIttaragEdlmanCYrQrbEnßmsRmab;plitpl
fµIenH rYcsnñidæankrNIEdleKRtUvbgçMcitþplitvacMnYn 2 Ékta.
x.etIR)ak;cMeNjmYyÉktaénplitplfµImantémøb:unµanÉktaeTIbeKRtUvbþÚrkarseRmccitþ
enH?
K. ]bmaR)ak;cMeNjmYyÉktaénplitplfµIKW $22 cUrrkcemøIyRbesIrbMput.
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
8
4. Rkumh‘unplitrfynþ TOYOTA plitrfynþBIrRbePTKW rfynþeTscr nigrfynþdwkTMnij.
Rkumh‘unmansgVak;plitkmµBIrlMdab;KW dMeLIg nig )aj;fñaM. kñúgmYyéf¶² eKmaneBlevlabRmug
sRmab;dMeLIg 250h nigeBlevlabRmugsRmab;)aj;fñaM 300h. kñúgmYyem:ag² eKdMeLIg)anrfynþ
eTscrcMnYnbYn b¤rfynþdwkTMnijcMnYnBIr ehIycMeBaHrfynþeTscrmYy eKRtÚvkareBl 30 naTI nig
cMeBaHrfynþdwkTMnij eKRtÚvkareBl 20 naTIsRmab;kar)aj;fñaM. eKrk)anR)ak;cMeNj $150 BIkar
lk;rfynþeTscrmYy nig $200 BIkarlk;rfynþdwkTMnijmYy.
k. etIkñúgmYyéf¶² Rkumh‘unRtÚvplitrfynþeTscr nigrfynþdwkTMnijcMnYnb:unµanedIm,I[ cMeNjGtibrma? (sresrcMeNaTenHCaragsþg;da rYcedaHRsayvatamviFIRkahiVk bnÞab;mkedaH
RsayvatamviFIsuImepøc).
x. Ep¥kelItRmÚvkarTIpSarEpñk Marketing énRkumh‘un)anpþl;B½t’manmkfa ebIRkumh‘un
plitrfynþ Sport enaHRkumh‘unnwgrk)anR)ak;cMeNj $300 kñúgrfynþmYy. eKdwgfa rfynþ
Sport mYyRtÚvkareBl 40 naTIsRmab;dMeLIg nigry³eBl 30 naTIsRmab;)aj;fñaM. etIRkumh‘unRtÚv
seRmccitþplitrfynþ Sport Edrb¤eT?
K. ebIRkumh‘unRtÚvseRmccitþplitrfynþ Sport enaH etIRkumh‘unRtÚvplitrfynþeTscr rf
ynþdwkTMnij nigrfynþ Sport cMnYnb:unµanedIm,I[R)ak;cMeNjGtibrma?
♥
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
9
CMBUkT I2 cMeNaTeTVPaB (Dual Problem)
I- cMeNaTd MbUg n igcMeNaTeTVPaB ( The Primal and Dual Problems )
RKb;cMeNaTénkarbegáItkmµviFIlIenEG‘r GacmancMeNaTkmµviFIlIenEG‘rmYyeTotBak;B½n§
CamYyvaEdleKehAfa cMeNaTeTVPaBrbs;va.
cMeNaTeTVPaBmanpÞúkB½t’manesdækic© EtgEtpþl;dl;GñkRKb;RKg ehIyvaEfmTaMgGac
eFVIkaredaHRsayTan;eBlevlaCagcMeNaTdMbUg ( Primal Problem ) kñúgxN³eBlEdlkarKNna
RtUv)ankat;bnßy.
CMhank ñúgkarbegáIteT VPaB ( Steps to Forms a Dual )
1- ebIsinCacMeNaTdMbUg KWCaGtibrmakmµ enaHcMeNaTeTVPaB KWCaGb,brmakmµ ehIypÞúy
mkvij .
2- témøenAEpñkxagsþaM (RHS) énlkçxNÐkMNt;rbs;cMeNaTdMbUgkøayeTAnwgCaemKuNén
GnuKmn_eKalbMNgrbs;cMeNaTeTVPaB.
3- emKuNénGnuKmn_eKalbMNgrbs;cMeNaTdMbUg nwgkøayeTACatémøenAEpñkxagsþaMénlkç
xNÐkMNt;rbs;cMeNaTeTVPaB.
4- karpøas;TIénemKuNlkçxNÐkMNt;rbs;cMeNaTdMbUg nwgkøayeTACaemKuNlkçxNÐkMNt;
rbs;cMeNaTeTVPaB.
5- cMeBaHvismIkarvij KWmansBaØapÞúymkvij.
karbegáItrUbmnþsRmab;cMeNaTeTVPaB ( Formulation of the Dual Problem )
k- GefrseRmccitþ ( The Decision Variables )
vismIkarlkçxNÐkMNt;nImYy²rbs;cMeNaTdMbUg kMNt;[manGefrseRmccitþmYykñúg
cMeNaTeTVPaB. eKGacniyayfa GefrseRmccitþtageday y1, y2, y3, ….. éncMeNaTeTVPaBman
cMnYnesµInwgcMnYnvismIkarlkçxNÐkMNt;éncMeNaTdMbUg.
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x- GnuKmn_eKalbMNg ( The Objective Function )
témøenAEpñkxagsþaMénlkçxNÐkMNt;rbs;cMeNaTdMbUg KWCaelxemKuNénGnuKmn_eKal
bMNgrbs;cMeNaTeTVPaB.
K- lkçxNÐkMNt; ( The Constraints )
vismIkarlkçxNÐkMNt;rbs;cMeNaTeTVPaBkMNt;dUceTA³
- cMnYnlkçxNÐkMNt;rbs;cMeNaTeTVPaB esµInwgcMnYnGefrseRmccitþrbs;cMeNaTdMbUg.
- vismIkarlkçxNÐkMNt;rbs;cMeNaTTaMgBIrmanTisedApÞúyKñaCanic©.
- elxemKuNerogKñaénGnuKmn_eKalbMNgrbs;cMeNaTdMbUg KWCatémø RHS éncMeNaT
eTVPaB.
- elxemKuNénvismIkarlkçxNÐkMNt;rbs;cMeNaTdMbUgenAtamCYrQrnImYy² KWCaelx
emKuNénvismIkarlkçxNÐkMNt;rbs;cMeNaTeTVPaBenAtamCYredkvij.
lkçN³sMxan;² ( The Major Properties )
cMeNaTdMbUg nigcMeNaTeTVPaBmanTMnak;TMngsMxan;²tamlkçN³dUcteTA³
1- ebIcMeNaTdMbUg CacMeNaTGtibrmakmµ enaHcMeNaTeTVPaBCacMeNaTGb,brmakmµ nig
Rcasmkvij.
2- cMeNaTeTVPaBmancemøIyGubTIm:al; luHRtaEtcMeNaTdMbUgmancemøIyGubTIm:al; nig
Rcasmkvij.
3- témøGnuKmn_eKalbMNgrbs;cMeNaTTaMgBIresµIKñaCanic©.
4- cMeNaTeTVPaBéncMeNaTeTVPaBmYy KWCacMeNaTdMbUgrbs;va. kñúgtaragcemøIycug
eRkayrbs;cMeNaTdMbUg eKGacTaj)ancemøIyrbs;cMeNaTeTVPaB nigRcasmkvij.
5- rvagcMeNaTTaMgBIr dMeNaHRsaytamviFIsIumepøcéncMeNaTNamYyGac[eKTajyk
cemøIyBItaragcugeRkayéncMeNaTmYyeTot)an.
kark MNt;ragsþg;daT UeTAéncMeNaTeTVPaB
( General Standard of the Dual Problems )
\LÚveyIgbegáItcMeNaTdMbUg CacMeNaTGtibrmakmµTUeTAmYyedaysresrCaTRmg;
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sþg;dadUcteTA³
rktémøGtibrmaén Z = c1x1 + c2 x2 + ....... + cj xj +....... + cn xn ( 1 )
epÞógpÞat;lkçxNÐkMNt;lIenEG‘r
a11x1 + a12x2 + ....... + a1jxj + ........... + a1nxn ≤ b1 a21x1 + a22x2 + ....... + a2jxj + ........... + a2nxn ≤ b2 ................................................................................ ................................................................................ ai1x1 + ai2x2 + ....... + aijxj + .............+ ainxn ≤ bi ( 2 ) ................................................................................ ................................................................................ am1x1 + am2x2 + ....... + amjxj + ........... + amnxn ≤ bm
nig
xj ≥ 0 ( 1 ≤ j ≤ n ) ( 3 )
Edl bi cj nig aij ( 1 ≤ i ≤ m, 1 ≤ j ≤ n ) CacMnYnBitefr ehIy xj CacMnYnBitminGviC¢man
EdlRtÚvkMNt;.
cMeNaTeTVPaBrbs;vaKW ³
rktémøGb,brmaén G = b1y1 + b2 y2 + ....... + bi yi +....... + bm ym ( 4 )
epÞógpÞat;lkçxNÐkMNt;lIenEG‘r
a11y1 + a21y2 + ....... + ai1yi + ........... + am1ym ≤ c1 a12y1 + a22y2 + ....... + ai2yi + ........... + am2ym ≤ c2 ................................................................................ ................................................................................ a1jy1 + a2jy2 + ....... + aijyi + .............+ amjym ≤ cj ( 5 ) ................................................................................ ................................................................................ a1ny1 + a2ny2 + ...... + ainyi + ........... + amnym ≤ cn
nig
yi ≥ 0 ( 1 ≤ i ≤ m ) ( 6 )
eyIgGacsegçbnUvkarkMNt;sresrenH CaTRmg;ma:RTIs³
rktémøGtibrmaén z = C X ( 7 )
Edlmanl½kçx½NÐkMNt;
A X ≤ b ( 8 )
nig X ≥ O ( 9 )
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Edl C CaviucT½rCYredkEdlman n-vimaRt A Cama:RTIsmanlMdab; m × n X CaviucT½rCYrQrEdl
man n-vimaRt nig b CaviucT½rCYrQrEdlman m-vimaRt ehIyeKkMNt;sresrCaTRmg;ma:RTIsKW
[ ]ncccC 21= , 11 12 1n 1
21 22 2n 2
m1 m2 mn n
a a a xa a a x
A , X
a a a x
= =
nig b
=
mb
bb
2
1
. vismIkarviucT½r X ≥ O
mann½yfa kMub:Usg;nImYy²én X CacMnYnminGviC¢man.
cMeNaTeTVPaBrbs;vaKW³
rktémøGb,brmaén G = b Y ( 10 )
Edlmanl½kçx½NÐkMNt;
AT Y ≥ C ( 11 )
nig Y ≥ O ( 12 )
]TahrN_TI1
cMeNaTdMbUg cMeNaTeTVPaB
Maximize profit Z = 300x1 + 250x2 Minimize cost G = 40y1 + 42y2 + 12y3 Subject to 2x1 + x2 ≤ 40 Subject to 2y1 + y2 + y3 ≥ 300 x1 + 3x2 ≤ 42 y1 + 3y2 ≥ 250 x1 ≤ 12 and y1, y2, y3 ≥ 0. and x1, x2 ≥ 0.
]TahrN_TI2
cMeNaTdMbUg cMeNaTeTVPaB
Minimize cost Z = 3x1 + 4x2 + 2x3 Maximize profit G = 40y1 + 50y2 + 30y3 Subject to x1 + 2x2 + 3x3 ≥ 40 Subject to y1 + 3y2 ≤ 3 3x1 + 5x2 + x3 ≥ 50 2y1 + 5y2 + 3y3 ≤ 4 3x2 + 2x3 ≥ 30 3y1 + y2 + 2y3 ≤ 2 and x1, x2, x3 ≥ 0. and y1, y2, y3 ≥ 0.
II- cemøIyRbesIrbMputéncMeNaTeTVPaB
(The Optimal Solution of the Dual Problems)
]TahrN_TI3³ eK[cMeNaTdMbUg
Maximize profit Z = 4x1 + 6x2 + 10x3 + 12x4 Subject to x1 + 3x2 + 2x3 + 4x4 ≤ 5
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x1 + x2 + 5x3 + 3x4 ≤ 15 x1, x2, x3, x4 ≥ 0.
cUrbegáItcMeNaTeTVPaB nigedaHRsaytamviFIRkahVik rYcTajrkcemøIyGubTIm:al;éncMeNaT
dMbUg.
tag iy CaGefrseRmccitþéncMeNaTeTVPaBRtÚvKñanwgvismIkarlkçxNÐkMNt;TI i én
cMeNaTdMbUg. dMeNaHRsaytamviFIRkahVik eyIg)ancemøIyRbesIrbMputéncMeNaTeTVPaBKW³
y1 = 5, y2 = 0, t1 = 1, t2 = 9, t3 = 0, t4 = 8 nig Min G = 25 .
BIcemøIyenH eyIgTaj)ancemøIyRbesIrbMputéncMeNaTdMbUgKW³
x1 = 0, x2 = 0, x3 = 5/2, x4 = 0, S1 = 0, S2 = 5/2 nig Max Z = 25 .
enAeBlEdleK)anedaHRsaycMeNaTdMbUgedayviFIsIumepøcrYc ehIyeKGacGancemøIy
éncMeNaTeTVPaBelItaragsIumepøccugeRkayenaHedaypÞal;Etmþg. eKkMNt;cemøIytamviFIdUcteTA³
- GancemøIyenAelI OBJ row .
- témøelxkñúgCYrQr x1, x2, x3, ……KWCatémøénGefrelIs t1, t2, t3,……. rbs;
cMeNaTeTVPaB.
- témøelxkñúgCYrQr s1, s2, s3, ……KWCatémøénGefrseRmccitþ y1, y2, y3,……. rbs;
cMeNaTeTVPaB.
]TahrN_TI4³ eK[cMeNaTdMbUg
Maximize profit z = 3x1 + 4x2
Subject to 0.3x1 + 0.5x2 ≤ 300
x1 + 1.5x2 ≤ 750
x1, x2 ≥ 0
dMeNaHRsaytamviFIsIumepøc[taragcemøIycugeRkay³
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__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
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Final Table Decision variables Slack variables
Basic x1 x2 S1 S2 S3 z RHS
S1
x2
X1
0
0
1
0
1
0
1
0
0
– 0.4
2
– 2
0.1
– 2
3
0
0
0
60
300
300
OBJ 0 0 0 2 1 1 2100
t1 t2 y1 y2 y3 G Surplus variables Decision variables
cMeNaTeTVPaBrbs;vakMNt;eday³ Minimize cost G = 300 y1 + 750 y2 + 600 y3 Subject to 0.3 y1 + y2 + y3 ≥ 3 0.5 y1 + 1.5 y2 + y3 ≥ 4 y1, y2, y3 ≥ 0
TajecjBItaragsIumepøccugeRkayéncMeNaTdMbUg eyIgTTYl)ancemøIyGubTIm:al;én
cMeNaTeTVPaBKW³ y1 = 0, y2 = 2, y3 = 1, t1 = 0, t2 = 0 nig Min G = 2100 .
]TahrN_TI5³ eK[cMeNaTdMbUg
Maximize Z = 5 x1 + 3 x2 + 4 x3 + x4 subject to 2 x1 + x2 + 3x3 + x4 ≤ 30 x1 + x3 + 2x4 ≤ 16 3x1 + 4x2 + 2x3 ≤ 36 x1, x2, x3, x4 ≥ 0
cUrbegáItcMeNaTeTVPaB nigedaHRsaytamviFIsaRsþsIumepøc rYcTajrkcemøIyGubTIm:al;én
cMeNaTdMbUg.
cMeNaTeTVPaBéncMeNaTdMbUgkMNt;eday³
Miniimize G = 30 y1 + 16 y2 + 36 y3 Subject to 2 y1 + y2 + 3y3 ≥ 5 y1 + 4y3 ≥ 3 3y1 + y2 + 2y3 ≥ 4 y1 + 2y2 ≥ 1 y1, y2, y3 ≥ 0
tamkmµviFI Lindo eyIgTTYl)ancemøIyGubTIm:al; nigtaragsIumepøccugeRkay³
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LP OPTIMUM FOUND AT STEP 3 OBJECTIVE FUNCTION VALUE 1) 63.33333 VARIABLE VALUE REDUCED COST Y1 0.333333 0.000000 Y2 0.333333 0.000000 Y3 1.333333 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 -10.222222 3) 2.666667 0.000000 4) 0.000000 -2.666667 5) 0.000000 -1.555556 NO. ITERATIONS= 3 THE TABLEAU ROW (BASIS) Y1 Y2 Y3 SLK 2 SLK 3 SLK 4 1 ART 0.000 0.000 0.000 10.222 0.000 2.667 2 Y2 0.000 1.000 0.000 -0.222 0.000 0.333 3 SLK 3 0.000 0.000 0.000 -1.778 1.000 0.667 4 Y3 0.000 0.000 1.000 -0.556 0.000 0.333 5 Y1 1.000 0.000 0.000 0.444 0.000 -0.667 ROW SLK 5 1 1.556 -63.333 2 -0.556 0.333 3 0.556 2.667 4 0.111 1.333 5 0.111 0.333
cemøIyrbs;cMeNaTdMbUg LP OPTIMUM FOUND AT STEP 3 OBJECTIVE FUNCTION VALUE 1) 63.33333 VARIABLE VALUE REDUCED COST X1 10.222222 0.000000 X2 0.000000 2.666667 X3 2.666667 0.000000 X4 1.555556 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 0.333333 3) 0.000000 0.333333 4) 0.000000 1.333333 NO. ITERATIONS= 3
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lMhat ;
1. begáItcMeNaTeTVPaBéncMeNaTdMbUgxageRkam³ Maximize z = 4x1 + 12x2 + 25x3
Subject to 3x1 + 4x2 + 7x3 ≤ 20 4x1 + 8x2 + 9x3 ≤ 30 7x1 + 6x2 + 2x3 ≤ 40
and x1, x2, x3 ≥ 0.
2. begáItcMeNaTeTVPaBéncMeNaTdMbUgxageRkam³ Minimize z = 23x1 + 41x2 + 30x3 + 60x4 + 70x5
Subject to 2x1 + 3x2 + 5x3 − 8x4 + 2x5 ≥ 80 3x2 + 2x3 + 7x4 + 8x5 ≥ 30
x1 + 6x2 + 2x3 + 6x4 + 5x5 ≥ 10 and x1, x2, x3, x4, x5 ≥ 0.
3. eK[cMeNaTdMbUgdUcxageRkam³ Minimize cost z = 10 x1 + 12 x2 Subject to x1 + x2 ≥ 3 x1 + 4x2 ≥ 2 and x1, x2 ≥ 0.
k. edaHRsaycMeNaTdMbUgtamviFIRkahVik.
x. cUrbegáItcMeNaTeTVPaB nigedaHRsaytamviFIsIumepøc rYcTajrkcemøIyéncMeNaTdMbUg.
4. eK[cMeNaTdMbUgdUcxageRkam³ Minimize cost z = 3 x1 + 12 x2 Subject to x1 + x2 ≥ 120 x1 + 3x2 ≥ 100 and x1, x2 ≥ 0.
k. edaHRsaycMeNaTdMbUgtamviFIRkahVik.
x. begáItcMeNaTeTVPaBnigedaHRsayvatamviFIsIumepøc rYcTajrkcemøIyéncMeNaTTaMgBIr.
5. eK[cMeNaTdMbUgdUcxageRkam³ Minimize cost z = 24 x1 + 30 x2 Subject to 2x1 + 3x2 ≥ 10 4x1 + 9x2 ≥ 15 6x1 + 6x2 ≥ 20 and x1, x2 ≥ 0.
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k. edaHRsaycMeNaTdMbUgtamviFIRkahVik.
x. begáItcMeNaTeTVPaBnigedaHRsayvatamviFIsIumepøc rYcTajrkcemøIyéncMeNaTTaMgBIr.
6. eK[cMeNaTdMbUgdUcxageRkam³ Maximize profit P = 80 x1 + 75 x2 Subject to x1 + 3x2 ≤ 4 2x1 + 5x2 ≤ 8 x1, x2 ≥ 0
k. edaHRsaycMeNaTdMbUgtamviFIRkahVik.
x. cUrbegáItcMeNaTeTVPaB nigedaHRsaytamviFIsIumepøc rYcTajrkcemøIyéncMeNaTdMbUg.
K. cUrkMNt;rkcMeNaTeTVPaBéncMeNaTeTVPaBrbs;cMeNaTdMbUgenH.
7. eK[cMeNaTeTVPaBdUcxageRkam³ Minimize cost G = 28 y1 + 53 y2 + 70 y3 + 18 y4 Subject to y1 + y4 ≥ 10 y1 + 2y2 + y3 ≥ 5 – 2y2 + y4 ≥ 31 5y3 ≥ 28 12y1 + 2y3 – y4 ≥ 17 and y1, y2, y3, y4 ≥ 0.
k. cUrbegáItTRmg;edIméncMeNaTdMbUg.
x. edayeRbIkmµviFI LINDO cUrbgðajtaragsIumepøccugeRkayéncMeNaTdMbUg rYcTajrk
cemøIyGubTIm:al;éncMeNaTTaMgBIr.
8. eK[cMeNaTdMbUg³ Maximize z = 2x1 + 3x2 + 2x3 + 4x4
Subject to x1 + 3x2 + 2x3 + 4x4 ≤ 60 3x1 + x2 + 2x3 + 3x4 ≤ 30 and x1, x2, x3, x4 ≥ 0.
begáItcMeNaTeTVPaB nigedaHRsayvatamviFIRkahVik rYcTajrkcemøIyGubTIm:al;éncMeNaT
eTVPaB nigcMeNaTdMbUg.
9. eK[cMeNaTdMbUg³ Minimize C = x1 + 4x2 + 2x3
Subject to x1 − 2x2 + 3x3 ≥ 40
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x1 + 3x2 − 2x3 ≥ 20 and x1, x2, x3 ≥ 0.
begáItcMeNaTeTVPaB nigedaHRsayvatamviFIRkahVik rYcTajrkcemøIyGubTIm:al;éncMeNaT
eTVPaB nigcMeNaTdMbUg.
10. eK[cMeNaTdMbUg³ Minimize z = 3x1 + 4x2 + 3x3 + 6x4 + 7x5
Subject to 3x1 + x2 + 2x3 − 3x4 + 2x5 ≥ 84 3x2 + x3 + 4x4 + 5x5 ≥ 30 and x1, x2, x3, x4, x5 ≥ 0.
begáItcMeNaTeTVPaB nigedaHRsayvatamviFIRkahVik rYcTajrkcemøIyGubTIm:al;éncMeNaT
eTVPaB nigcMeNaTdMbUg.
11. cUreRbIkmµviFI LINDO rbs;GñkedIm,IedaHRsaycMeNaTxageRkam CadMbUgedaHRsaycMeNaTeTV
PaB ehIybnÞab;mkedaHRsaycMeNaTdMbUg³
Minimize cost C = 5 x + 6 y + 3 z subject to 5 x + 5 y + 3 z ≥ 50 x + y – z ≥ 20 7x + 6y – 9z ≥ 30 5x + 5y + 5z ≥ 35 2x + 4y – 15z ≥ 10 12x + 10y ≥ 90 y – 10z ≥ 20 and x, y, z ≥ 0.
♥
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CMBUkT I3 karv iPaKeRkay cemøIy RbesIrbMput (Postoptimality Analysis)
I- esckþIepþIm (Introduction)
kñúgBiPBesdækic©GVI²TaMgGs;EtgEtmanbERmbRmÜldUcCa
- vtßúFatuedImTaMgLayGacmaneRcIneLIg² b¤k¾GacxSt;eTA².
-tRmÚvkarplitplmYyGacekIneLIg b¤fycuH.
-kRmitR)ak;cMeNjGacekIneLIg b¤fycuH.
plitkrRbQmmuxnwgcMeNaTmYy faetIKeRmagplitkmµd¾RbesIrbMputBImunvaenAEtRbesIrbMput
dEdlb¤eT eBlEdlmanlkçxNÐfµIERbRbÜlxusBImun. edIm,Iyl;dwgnUvbBaðaenH CaFmµtaeyIgRtÚveFVI
karviPaKnUvplb:HBal;mYyeRkayBIrkeXIjcemøIyRbesIrbMputéncMeNaT.
II- karv iPaKGgÁxagsþaM (RHS Analysis)
]TahrN_TI1³ eK[cMeNaTkmµviFIlIenEG‘rdUcxageRkam
Maximize z = 3x1 + 4x2
Subject to 0.3x1 + 0.5x2 ≤ 300 (1)
x1 + 1.5x2 ≤ 750 (2)
x1 + x2 ≤ 600 (3)
and x1, x2 ≥ 0.
k- edaHRsaycMeNaTxagelIenHtamviFIRkahVik.
x- edaHRsaycMeNaTxagelIenHtamviFIsuImepøc.
K- sikSabERmbRmÜlkñúgka)a:sIuFI (1), (2) nig (3).
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cMeBaHsMNYr x- tamviFIsuImepøc eyIg)antaragsuImepøccugeRkay³
Basic x1 x2 S1 S2 S3 z RHS
S1
x2
x1
0
0
1
0
1
0
1
0
0
– 2/5
2
– 2
1/10
– 2
3
0
0
0
60
300
300
OBJ 0 0 0 2 1 1 2100
III- karv iPaKR)ak;cMeNjÉkta (Unit Profit Analysis)
]TahrN_TI2³ eK[cMeNaTkmµviFIlIenEG‘rdUcxageRkam
Maximize z = 300x1 + 150x2
Subject to 2x1 + x2 ≤ 40 (1)
x1 + 3x2 ≤ 60 (2)
3x1 + x2 ≤ 30 (3)
and x1, x2 ≥ 0.
k- edaHRsaycMeNaTxagelIenHtamviFIRkahVik.
x- edaHRsaycMeNaTxagelIenHtamviFIsuImepøc.
K- sikSabERmbRmÜlemKuN x1 nig x2énGnuKmn_eKalbMNg z . cMeBaHsMNYr x- tamviFIsuImepøc eyIg)antaragsuImepøccugeRkay³
Basic z x1 x2 S1 S2 S3 RHS
S1
x2
x1
0
0
0
0
0
1
0
1
0
1
0
0
– 1/8
3/8
– 1/8
1/24
– 1/8
3/8
110/8
150/8
30/8
OBJ 1 0 0 0 150/8 750/8 25500/8
♣ ♣♣♣♣
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lMhat ;
1. cMeNaTkmµviFIlIenEG‘rmYymanGnuKmn_eKalbMNg z = $150x1 + $200x2 EdlRtUvrktémø
Gtibrma. eK)anedaHRsaytamviFIsuImepøc ehIyTTYl)antaragsIumepøccugeRkay³
Basic x1 x2 S1 S2 z RHS
x2
x1
0
1
1
0
3
– 2
– 3/2
3
0
0
300
400
OBJ 0 0 300 150 1 120 000
k- sikSaplb:HBal; nigbERmbRmÜlrbs; S1 nig S2 tamtaragxagelI rYcepÞógpÞat;eday
smIkar rYcrklImIténka)a:suIFI.
x- sikSabERmbRmÜlemKuN x1 nig x2énGnuKmn_eKalbMNg z tamtaragxagelI.
2. eK[cMeNaT LP Edlmanragsþg;dadUcxageRkam³ Maximize z = 32x1 + 18x2 + 19x3
Subject to 2x1 + x2 + x3 ≤ 12 8x1 + 10x2 + 3x3 ≤ 90
4x1 + 4x2 + 3x3 ≤ 30 and x1, x2, x3 ≥ 0.
k- edaHRsaycMeNaTxagelIenHtamviFIsuImepøc.
x- viPaKeRkaycemøIyRbesIrbMputelIGgÁxagsþaMtamtaragsIumepøccugeRkay.
K- viPaKeRkaycemøIyRbesIrbMputelIemKuNénGnuKmn_eKalbMNg z tamtaragsIumepøc
cugeRkay.
X- ]bmafa eKRtÚvbgçMcitþplit x2 cMnYn 3Ékta. rkplb:HBal;elIplitkmµ x1 nig x3
nigR)ak;cMeNj.
3. eK[cMeNaTkmµviFIlIenEG‘r ( LPP ) manragsþg;dadUcxageRkam³ Maximize 1 2z 300x 150x= +
subject to 1 22x x 40 (1)+ ≤
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
22
1 2x 3x 60 (2)+ ≤
1 23x x 30 (3)+ ≤ and 1x 0≥ , 2x 0≥ .
k- sresrkUdkñúgkmµviFI LINDO .
x- bgðajnUvtaragsuImepøcdMbUgtamkmµviFI LINDO .
K- bgðajnUvlT§plBIkmµviFI LINDO .
X- bgðajnUvtaragsuImepøccugeRkaytamkmµviFI LINDO .
g- viPaKeRkaycemøIyRbesIrbMputelIGgÁxagsþaM nigelIemKuNénGnuKmn_eKalbMNg z
tamsMNYr K- .
4. Rkumh‘un Darling Downs Tinpots pliteRKÓgtubEtgpÞHEdlxøÜnGaclk;ecj)anbrimaNeRcIn
minkMNt;edaytémøbc©úb,nñrbs;va. eRKÓgtubEtgmanbIRbePTKWRbePT A B nig C RtUv)anplit
kalBIeBlmun Etbc©uúb,nñenH eKkMBugeFVIkarvaytémøplitkmµenaHeLIgvij. Rkumh‘unmanmnusS
CMnaj 5 nak;eFVIkar)an 40h kñúgmYys)aþh_ ehIyCaRbcaMs)aþh_m:asIunGaceRbIR)as;)anGs;ry³
eBl 45h . buKÁlikTaMgGs;GacTTYleFVIkic©karTaMgGs;edayel,ÓnswgEtdUcKña dUcenHvaKµanbBaða
kñúgkarerobcMkalviPaKkargareT. GMLúgeBlkargarRtUvkarsRmab;eRKÓgtubEtgnImYy² ehIynig
lT§plénR)ak;cMeNjRtUv)anbgðajkñúgtaragxageRkam³
Ornament A B C Man hours required ½ 1/3 2/5 Machine hours required 1/10 1/12 3/20 Profit ( $ ) 6.00 4.20 5.00
Rkumh‘uncg;begáInR)ak;cMeNjrbs;xøÜn[mankRmitGtibrma eday)anrkSanUvlkçxNÐkMNt;
énBlkmµ niglT§PaBénm:asIun.
k. cUrbegáItTRmg;sþg;daéncMeNaT ehIyedaHRsayvaedaysMeNr nigedayeRbIkmµviFI
Lindo rbs;Gñk.
x. cUreRbItaragsuImepøccugeRkayrbs;Gñk (minyktamTinñn½yénlT§plrbs;kMuBü ÚT½reT)
edIm,IeqøIysMNYrminTak;TgKñadUcteTA³
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
23
(i). etImUlehtuGVI)anCaR)ak;cMeNjelIeRKÓgtubEtgRbePT C RtUvEteLIgelIs 64 cents
muneBlRkumh‘unGaccMeNj)anedaykarpliteRKÓgtubEtgRbePT C ?
(ii). ebIR)ak;cMeNjelIplitpl C ekIndl; $6.20 ehIyR)ak;cMeNjelIplitpl B fy
cuHmkRtwm $4 etIkalviPaKplitkmµKYrEteTACay:agNa?
(iii).etIEdntémøénR)ak;cMeNjelIplitpl A esµIb:unµancMeBaHcemøIyGubTIm:al;kñúgsMNYrk.?
(iv). kmµkrxøHsµ½RKcitþeFVIkarelIsem:agedayTTYlR)ak;bEnßm $10 kñúgmYyem:ag. etIRkum
h‘unKYrTTYlsMeNIenaHEdrb¤eT? ebITTYl etIkmµkrKYrEteFVIkarelIsb:unµanem:ag? ebIminTTYl etIman
mUlehtuGVI?
(v). ebIeKCYlkmµkrmñak;bEnßm etIkalviPaKplitkmµnwgpøas;bþÚry:agNa?
(vi). s)þah_enH cMnYnem:agEdlm:asuIneRbIR)as;fycuHGs; 3h BIeRBaHEttRmUvkarCYsCul
m:asIun. etIRkumh‘unnwgxatGs;R)ak;b:unµanBIkar)at;bg;eBlevlaenH ehIysRmab;s)aþh_enHEdr
plitkmµRbcaMs)aþh_nwgKYréleTACay:agNacMeBaHR)ak;cMeNjGtibrma?
(vii).Rkumh‘unxagelIseRmccitþfa KYrEtplity:agehacNas;eRKÓgtubEtgRbePT C cMnYn
50 sRmab;s)aþh_nImYy². etIplitkmµRbcaMs)aþh_KYrEtERbeTACay:agNa?
(viii). Rkumh‘uneTscrN_cg;[Rkumh‘un Darling Downs Tinpots plitEkvFM² EdleK
rMBwgfa eRbIry³eBlBlkmµ 15 mn nigry³eBlma:sIun 7 mn edIm,InwgplitEkvmYy. RkumRbwkSa
eTscrN_esñITijEkvFMTaMgenHenAtémømYyEdlpþl;[Rkumh‘unnUvR)ak;cMeNj $4.8 BIEkvnImYy².
etIRbtikmµrbs;Rkumh‘unKYrEty:agNacMeBaHesckþIesñIenaH?
♥
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
24
CMBUkT I4 cMeNaTkardwkCBa¢Ún (Transportation Problems)
III- esckþIepþIm (Introduction)
m:UEdlkardwkCBa¢Ún KWCam:UEdlEdleKGacykeTAedaHRsay edayeRbIR)as;rebob
KNnaEdlmanRbsiT§PaBCagebIeRbobeFobeTAnwgviFIsIumepøc. cMeNaTkardwkCBa¢Ún KWCaEpñkmYy
énviFIkmµviFIlIenEG‘r EdleKehAfa cMeNaTlMhUrbNþaj (Network Flow Problems). cMeNaT
enHRtUvkMNt;GMBIKeRmageBlénkardwkCBa¢ÚnsRmab;eragcRk]sSahkmµ EdlRtUveFVIy:agNa[éfø
dwkCBa¢ÚnsrubmantémøGb,brma.
cMeNaTdwkCBa¢Ún RtUv)anGnuvtþeTAelIkarEbgEckTMnijBIkEnøgmYycMnYnénkarpÁt;pÁg;
( TIRbPB ) eTATIkEnøgdéTeToténtRmUvkar ( eKaledA ) . ebImancMeNaTénkardwkCBa¢ÚnTUeTAmYy
enaHeyIgGacdwg)anBIbBaðadUcCa³
RbPB ( Sources ) : CacMNucedImdUcCa eragcRk GNþÚgEr: b¤sßab½nnanaEdlbegáItnUv
smÖar³pÁt;pÁg;énFnFanmankMNt;dUcCa plitpl b¤vtßúFatuedIm.
eKaledA ( Sink ) : CacMNuccugeRkay b¤TisedAdUcCa XøaMgsþúkTMnij cMNt b¤haglk;
dUrEdlRtUvkar b¤eRbIR)as;nUvFnFan.
smÖar³pÁt;pÁg; ( Supply ) : CabrimaN b¤smtßPaBénFnFanmankMNt;eTAtamRbPB
nImYy².
tRmUvkar ( Demand ) : CaesckþIRtUvkarnUvRbPBFnFanenAtameKaledAnImYy².
pøÚv ( Pathways ) : karepÞrFnFanEdleKGnuBaØat[eFVIBIRbPBeTAkan;eKaledA.
éfødwkCBa¢ÚnkñúgmYyÉkta ( Per unit shipping cost ) : CaéføsRmab;dwkCBa¢ÚnTMnij b¤vtßú
mYyÉktadUcCamYyetan plitplmYy cMNuHmYyLanEdldwkBIRbPBc,as;las;mYyeTAkan;eKal
edACak;lak;mYy.
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
25
eKalbMNgéncMeNaTenH KWedIm,IerobcMkMNt;kardwkCBa¢ÚnBIcMNucRbPBeTAkan;cMNuc
eKaledA dUecñHéføedImsrubénplitpl nigkardwkCBa¢ÚnRtUv)aneKkat;bnßy[)anCaGtibrma.
]TahrN_TI1 cMeNaTénkardwkCBa¢Ún Factories Warehouses (Source) (Destination) 100 unis 300 unis 300 unis 200 units 300 unis 200 units Capacities Shipping Routes Requirements
]TahrN_TI2 cMeNaTénkardwkCBa¢Ún
]sSahkrmñak;EdlplitExSTUrsBÞ maneragcRkplitBIrkEnøgmYytaMgenA Salt Lake City
nigmYyeTotenA Denver ehIynigmanXøaMgsþúkTMnijcMnYnbIkEnøgeTotEdlmYyenA Los Angeles
mYyenA Chicago nigmYyeTotenA New York City. tRmUvkarénXøaMgsþúknImYy²KitCaetan
plitplpÁt;pÁg;EdlGacpÁt;pÁg;[eragcRknImYy²)anKitCaetan nigéfødwkCBa¢Únénplitplkñúg
mYyetan²KitCaduløarRtUv)anbgðajkñúgtaragTI1xageRkam³
taragTI1
From To Los Angeles Chicago New York City Supplies
Salt Lake City 5 7 9 100
Denver 6 7 10 140
Demand 100 60 80
etIeKKYrdwkCBa¢ÚnExSTUrsBÞcMnYnb:unµanetanBIeragcRknImYy²eTA[XøaMgnImYy² edIm,I[éfø
dwkCBa¢ÚnsrubmantémøGb,brma ehIyvabMeBjesckþIRtUvkarpgEdr?
dMeNaHRsay
F1 W1
F2 W2
F3 W3
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
26
tag xij CacMnYnKitCaetanénExSTUrsBÞdwkBIeragcRk i eTAXøaMg j cMeBaH i = 1, 2 ( 1 = Salt
Lake City, 2 = Denver) nigcMeBaH j = 1, 2, 3 ( 1 = Los Angeles, 2 = Chicago, 3 = New
York City). GnuKmn_eKalbMNg KWrktémøGb,brmaénéfødwkCBa¢Únsrub C. cMeNaTxagelIGac
sresrCaTRmg;KNitviTüa.
Minimize C = 5x11 + 7x12 + 9x13 + 6x21 + 7x22 + 10x23
Subject to x11 + x12 + x13 ≤ 100 (only 100 tonnes available at plant 1)
x21 + x22 + x23 ≤ 140 (only 140 tonnes available at plant 2)
x11 + x21 ≥ 100 (demand of 100 tonnes at warehouse 1
must be met)
x12 + x22 ≥ 60 (demand of 60 tonnes at warehouse 2 must be met)
x13 + x23 ≥ 80 (demand of 80 tonnes at warehouse 3 must be met)
xij ≥ 0 for all i, j (non-negativity constraints) Input In Lindo Min 5x11 + 7x12 + 9x13 + 6x21 + 7x22 + 10x23
Subject to x11 + x12 + x13 = 60 !demand of 60 tonnes at warehouse 2 must be met
x13 + x23 >= 80 !demand of 80 tonnes at warehouse 3 must be met
end
Output LP OPTIMUM FOUND AT STEP 4
OBJECTIVE FUNCTION VALUE
1) 1720.000
VARIABLE VALUE REDUCED COST
X11 100.000000 0.000000
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
27
X12 0.000000 1.000000
X13 0.000000 0.000000
X21 0.000000 0.000000
X22 60.000000 0.000000
X23 80.000000 0.000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 0.000000 1.000000
3) 0.000000 0.000000
4) 0.000000 -6.000000
5) 0.000000 -7.000000
6) 0.000000 -10.000000
NO. ITERATIONS= 4
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X11 5.000000 0.000000 6.000000
X12 7.000000 INFINITY 1.000000
X13 9.000000 1.000000 0.000000
X21 6.000000 INFINITY 0.000000
X22 7.000000 1.000000 7.000000
X23 10.000000 0.000000 1.000000
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 100.000000 80.000000 0.000000
3 140.000000 INFINITY 0.000000
4 100.000000 0.000000 80.000000
5 60.000000 0.000000 60.000000
6 80.000000 0.000000 80.000000
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
28
RbsinebIeyIgdak;lkçxNÐfa smÖar³pÁt;pÁg;RtUvEteRbI[Gs; ehIyRKb;tRmUvkarRtUvEt
bMeBj[)anCadac;xat mann½yfa enAeBlNaEdlbrimaNpÁt;pÁg;srubesµInwgtRmUvkarsrub enaH
eyIg)anrUbmnþéncMeNaTenHeTACa³
Minimize C = 5x11 + 7x12 + 9x13 + 6x21 + 7x22 + 10x23 Subject to x11 + x12 + x13 = 100
x21 + x22 + x23 = 140 x11 + x21 = 100
x12 + x22 = 60 x13 + x23 = 80
xij ≥ 0 for all i = 1, 2; j = 1, 2, 3 Output From Lindo OBJECTIVE FUNCTION VALUE
1) 1720.000
VARIABLE VALUE REDUCED COST
X11 20.000000 0.000000
X12 0.000000 1.000000
X13 80.000000 0.000000
X21 80.000000 0.000000
X22 60.000000 0.000000
X23 0.000000 0.000000
II- cemøIyd MbUgéncMeNaTkard wkCBa¢Ún
(Initial Solution of Transportation Problems)
eyIgGacrkeXIjcemøIyEdlGacyk)andMbUgmYyedayeRbIviFIBIry:agKW³ (1). viFIeRCIserIsRbGb;RCugxageCIg-lic
(2). viFIeRCIserIsRbGb;manéføTabCageK.
]TahrN_TI3
Rkumh‘unpliteRKÓgsgðarwm Executive Furniture Corporation EdleragcRkenHplittu
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
29
kariyal½ysßitenAtMbn;bIepSgKña KWtMbn; Des Moines Evansville nig Fort Lauderdale . Rkum
h‘unEbgEcktuEdlplitehIyeTAtamXøaMgenAtMbn;epSg²KñaEdlsßitenA Albuquerque Boston
nig Cleveland. kar)a:n;sµanbrimaNplitplRbcaMExenAtameragcRknImYy² nigkar)a:n;sµan
cMnYnéntuEdlRtUvkarRbcaMExenAtamXøaMgnImYy²RtUv)anbgðajenAkñúgtaragTI1.
Rkumh‘un)anKiteXIjfa éføedIménplitpltunImYy² vakMNt;eTAelIeragcRkerog²xøÜn b:uEnþ
éføedImEdlmankarERbRbYlenaH KWéfødwkCBa¢ÚnBIkEnøgRbPBnImYy²eTAkEnøgeKaledAnImYy². éfø
edImdwkCBa¢ÚnnwgmanbgðajkñúgtaragTI1xageRkam³
taragTI1
From To Albuquerque (A) Boston (B) Cleveland(C) Factory Capacity
Des Moines ( D ) $5
$4 $3 100
Evansville ( E ) $8
$4 $3 300
Fort Lauderdale ( F ) $9
$7 $5 300
Warehouse Demand 300 200 200 700
bBaðaEdlRtUvedaHRsayenaH KWfaetIeKRtUveRCIserIsykpøÚvNakñúgkardwkCBa¢Ún ehIynig cMnYntub:unµanEdlRtUvdwktampøÚvnImYy²edIm,IeFVIy:agNa[)anGb,brmakmµéfødwkCBa¢Únsrub.
dMeNaHRsay
(1). v iFIeRCIserIsRbGb;RCúgxageCIg∼lic (The North-West Corner Cell Method)
viFIeRCIserIsRbGb;RCugxageCIg-lic CaviFImYyEdlcab;epþImBIRbGb; (Cell) xagelI
bMput ehIyxageqVgbMputéntaragedaybMeBjnUvTinñn½ymYyesµInwg Min(Supply, Demand). viFI
enH eKGacrk)ancemøIydMbUgmYyedayRKan;EtbMeBjRbGb;enACYredkelIbMput nigeqVgbMputmuneK
rYceRbIviFandEdlenHedIm,IbMeBjRbGb;bnþbnÞab;eTot.
(a). eFVIkarpÁt;pÁg;[Gs;énbrimaNenARbPB (eragcRk) énCYredknImYy²munnwgpøas;TIcuH
eTAkan;CYredkbnÞab;eTot.
(b). bMeBj[RKb;nUvtRmUvkarenATIkEnøgpÁt;pÁg; (XøaMg) énCYrQrnImYy²munnwgeFVIkarpøas;
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
30
TIeTAkan;RbGb;xagsþaMenACYrQrbnÞab;.
(c). Binitüfa ral;karpÁt;pÁg;Gs;ehIyb¤enA nigtRmUvkarRKb;Gs;b¤eT.
taragTI2 From To Albuquerque (A) Boston (B) Cleveland(C) Factory Capacity
Des Moines ( D )
$5 100
$4 $3
100
Evansville ( E )
$8 200
$4 100
$3
300
Fort Lauderdale ( F ) $9 $7 100
$5 200
300
Warehouse Demand 300 200 200 700
eragcRk D pÁÁt;pÁg;Gs; RKb;tamtRmUvkarrbs;XøaMg A
eyIgGaceFVItaragsegçbénkarKNnaéføQñÜldwkCBa¢ÚndUcteTA³
taragTI3
Routes Units Shipped Per Unit Cost ($) Cost ($) D to A E to A E to B F to B F to C
100 200 100 100 200
5 8 4 7 5
500 1600 400 700
10000 Total Transportation Cost (TTC) 4200
(2). v iFIeRCIserIsRbGb;manéføTabCageK ( Least Cost Cell Method )
taragxagelIbgðajfa karpÁt;pÁg;srubrbs;eragcRkesµInwgtRmUvkarsrubrbs;XøaMg. eKGac
TTYl)ancemøIydMbUgmYy edayRKan;EtbMeBjRbGb;EdlmanéføQñÜlTabCageKCamunnUvTinñn½ymYy
esµInwg Min(Supply, Demand) rYcehIybMeBjRbGb;EdlmanéføQñÜlx
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
31
taragTI4
From To Albuquerque (A) Boston (B) Cleveland(C) Factory Capacity
Des Moines ( D ) $5
$4 $3 100
100
Evansville ( E ) $8 $4 200
$3 100
300
Fort Lauderdale ( F ) $9 300
$7 $5 300
Warehouse Demand 300 200 200 700
eyIgGaceFVItaragsegçbénkarKNnaéføQñÜldwkCBa¢ÚndUcteTA³
taragTI5
Routes Units Shipped Per Unit Cost ($) Cost ($)
D to C E to C
E to B F to A
100 100
200 300
3 3
4 9
300 300
800 2700
Total Transportation Cost (TTC) 4100
III- v iFIrktémøRbEhlv :UEhÁl (The Vogel’s Approximation Method)
\LÚvenH eyIgnwgbgðajviFIrktémøRbEhlvU:EhÁl EdlCaviFImYypþl;nUvcemøIydMbUgbMput
mYyEdlsßitenACitcemøIyRbesIrbMputCagcemøIydMbUgEdl)anedayeRbItamviFIeRCIserIsRbGb;RCug
xageCIg-lic b¤viFIeRCIserIsRbGb;manéføTabCageK.
eKrkcemøIydMbUgbMputtamviFIrktémøRbEhlvU:EhÁldUcxageRkam³
1-KNnaR)ak;Bin½y (Penalty): kñúgCYredk nigkñúgCYrQrnImYy² rkpldkrvagéfødwkCBa¢Ún
bnÞab;TabbMput nigéfødwkCBa¢ÚnTabbMputEdlbgðajGMBIKuNvibtþiénkareRbIR)as;RbGb; (Cell) Edl
mantémøTabbMput ehIyxkxanmin)aneRbIR)as;RbGb;EdlmantémøTabbnÞab; pldkenHehAfa
R)ak;Bin½y.
2-eRCIsykCYredk b¤CYrQrEdlmanR)ak;Bin½yx
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
32
3-KNnapldkrvagkarpÁt;pÁg; nigTinñn½ykñúgRbGb;enH ehIypldkrvagtRmÚvkar nigTinñ
n½ykñúgRbGb;enH.
- ebI karpÁt;pÁg; − Tinñn½ykñúgRbGb; = 0 nig tRmÚvkar − Tinñn½ykñúgRbGb; ≠ 0 enaHeKRtÚv
lubCYredkEdlRtÚvnwgRbGb;énRCúgxageCIg-lic rYcGnuvtþviFIxagelIcMeBaHtaragfµI.
- ebI karpÁt;pÁg; − Tinñn½ykñúgRbGb; ≠ 0 nig tRmÚvkar − Tinñn½ykñúgRbGb; = 0 enaHeKRtÚv
lubCYrQrEdlRtÚvnwgRbGb;énRCúgxageCIg-lic rYcGnuvtþviFIxagelIcMeBaHtaragfµI.
- ebI karpÁt;pÁg; − Tinñn½ykñúgRbGb; = 0 nig tRmÚvkar − Tinñn½ykñúgRbGb; = 0 enaHeKRtÚv
lubCYredk nigCYrQrEdlRtÚvnwgRbGb;énRCúgxageCIg-lic rYcGnuvtþviFIxagelIcMeBaHtaragfµI.
eKeFVIrebobenHCabnþbnÞab;rhUteK)ancemøIydMbUgbMputmYy.
taragTI6
From To Albuquerque (A) Boston (B) Cleveland(C) Supply Penalty 1
Des Moines (D) $5 $4 $3
100
Evansville (E) $8 $4
$3
300
Fort Lauderdale (F) $9 $7
$5
300
Demand 300 200 200 700 Penalty 1
IV- v iFIsþibPIgsþÚn ³ rkcemøIyéføed ImTabbMput
(Stepping-Stone Method: Finding a Least-Cost Solution) viFIsþibPIgsþÚn KWCaviFIsaRsþKNnaRcMEdl edIm,IeFVIy:agNa[cemøIydMbUgkøayeTACa
cemøIyGubTIm:al; (Optimal Solution) sRmab;dMeNaHRsaybBaðadwkCBa¢ÚnEdldMeNaHRsayenH
GacRbRBwtþeTA)anluHRtaEtcMnYnCYrQr nigcMnYnCYredkbUkbBa©ÚlKña ehIydknwgmYyesµInwgcMnYnpøÚv
EdlRtUvdwkCBa¢Ún (kñúgcMeNaTrbs;eyIgcMnYnpøÚvesµInwg 3 + 3 – 1 = 5).
CMhanTaMgR)aMkñúgkarsakl,gRbGb;mineRbIR)as;
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
33
(1). eRCIserIsRbGb;minTan;eRbIR)as;edIm,IvaytémøecjBItaragTI2 (cemøIydMbUgtamviFI
eRCIserIsRbGb;RCugxageCIg-lic) .
(2). cab;epþImnUvRbGb;EdleRCIserIsrYc ehIyKUscuc²ecjBIRbGb;Edl)aneRCIserIs
ehIypøas;bþÚrTItamTisQr nigedkedaycugRBYjRtUvsßitenAkñúgRbGb;NaEdlmanbrimaN.
(3). cab;epþImCamYysBaØabUk (+) enARbGb;EdlmineRbIR)as; ehIydak;sBaØaqøas; dk (–)
bUk (+) rhUtmkdl;RbGb;edImvij.
(4). KNnaenAkñúgRbGb;Eklm¥ (Improving Index) edaybEnßm 1 ÉktaeTAkñúgtarag
EdlmansBaØabUk (+) ehIydkecj 1 ÉktaBItaragEdlmansBaØadk (–) .
(5). eFVIeLIgvijmþgeTotBICMhanTI 1 dl;TI 4 rhUtdl;RbGb;RtUv)anEklm¥enaH RtUv)an
KNnacMeBaHRKb;RbGb;EdlmineRbIR)as;. ebIsinCaral;karKNnamantémøFMCag b¤esµIsUnü enaH
cemøIyGubTIm:al;RtUv)aneKrkeXIj. ebIsinCaminTan;rkeXIjeT enaHnwgRtUvEklm¥cemøIyenH
ehIynwgbnßyéfødwkCBa¢Únsrub. taragTI7
From To Albuquerque (A) Boston (B) Cleveland(C) Factory Capacity
Des Moines ( D )
$5 100 –
–
$4
+ Start
$3 + Start
100
Evansville ( E )
+ $8 200 + -
– $4 – 100 -
+
$3 + Start
300
Fort Lauderdale ( F )
$9
Start +
$7 + 100 +
-
– $5 - 200
300
Warehouse Demand 300 200 200 700
RbGb;EdlRtUvEklm¥KW
-KnøgEdleyIgeRbI³ + DB – DA + EA – EB D to B index = IDB = +(1 × $4) – (1 × $5) + (1 × $8) – (1 × $4) = + $3 -KnøgEdleyIgeRbI³ + DC – DA + EA – EB + FB – FC D to C index = IDC = + $3 – $5 + $8 – $4 + $7 – $5 = + $4
-KnøgEdleyIgeRbI³ + EC – EB + FB – FC E to C index = IEC = + $3 – $4 + $7 – $5 = + $1
-KnøgEdleyIgeRbI³ + FA – FB + EB – EA
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F to A index = IFA = + $9 – $7 + $4 – $8 = – $2
lT§plEdl)anmkBIkarEklm¥dMeNaHRsay
kñúgRbGb;sBaØadk (–) EdlRtUv)anKNnaedayviFIsþibPIgsþÚn tag[cMnYnsrubénéføQñÜl
dwkCBa¢ÚnEdlGacmanlT§PaBbBa©úHéføQñÜldwkCBa¢Ún ebIsinCamYy 1 Ékta b¤plitplRtUv)andwk.
eyIgrkeXIjRbGb;EtmYyKt;EdlmantémøGviC¢man –2 BIeragcRk F eTAXøaMg A . ebIsinCaman
RbGb;EdlmantémøGviC¢maneRcInCagmYy enaHkarEklm¥bnÞab; KWeRCIserIsykRbGb;mineRbIR)as;
NaEdlmantémøGviC¢manxøaMgCageK .
edaysarenARbGb;BI F eTA A mantémøGviC¢man enaHeyIgcab;epþImruHerIRbGb;BI F eTA A
muneK ehIyruHerIRbGb;CabnþbnÞab;EdlmancugRBYj b:uEnþeyIgminGacruHerIRbGb;EdlminmansBaØa
cugRBYj)aneT.
karEklm¥GacRtUv)aneFVIedaykardwkCBa¢ÚncMnYnEdlGacmanlT§PaBCaGtibrmaBI F eTA
A (emIltaragTI8xageRkam). edayRbGb;BI E eTA A nigBI F eTA B mansBaØadk (–) dUcenH
eyIgcab;epþImdk 100 ÉktaBIRbGb;BI E eTA A eTAdak;enARbGb;BI F eTA A muneK BIeRBaHRbGb;
enHmantémøGviC¢man (–$2) ehIykñúgRbGb;BI F eTA B RtUvdk 100 ÉktaeTAbUkbEnßmenAkñúg
RbGb;BI E eTA B BIeRBaHenAkñúgRbGb;enHenAxVH 100 ÉktaeToteTIbRKb; 200 ÉktatamtRmUvkar
rbs; B . taragTI8 ( Second Solution )
From To Albuquerque (A) Boston (B) Cleveland(C) Factory Capacity
Des Moines ( D ) $5 100 $4
$3
100
Evansville ( E )
$8 100
-
$4 200
+
$3 300
Fort Lauderdale ( F )
$9 100 +
$7
-
$5 200
300
Warehouse Demand 300 200 200 700
RbGb;EdlRtUvEklm¥KW
-KnøgEdleyIgeRbI³ + DB – DA + EA – EB D to B index = IDB = + $4 – $5 + $8 – $4 = + $3
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35
-KnøgEdleyIgeRbI³ + DC – DA + FA – FC D to C index = IDC = + $3 – $5 + $9 – $5 = + $2
-KnøgEdleyIgeRbI³ + EC – EA + FA – FC E to C index = IEC = + $3 – $8 + $9 – $5 = – $1
-KnøgEdleyIgeRbI³ + FB – EB + EA – FA F to B index = IFB = + $7 – $4 + $8 – $9 = + $2
tamry³karEklm¥dUcmun edayGnuvtþcMeBaHRbGb;BI E eTA C enaHeyIgTTYl)antaragTI9
EdlCataragcugeRkay.
taragTI9 (Third and Optimal Solution)
From To Albuquerque (A) Boston (B) Cleveland(C) Factory Capacity
Des Moines ( D ) $5 100 $4
$3
100
Evansville ( E )
$8 –
$4 200
Start $3 + 100
300
Fort Lauderdale ( F )
$9 200 +
$7
$5 – 100
300
Warehouse Demand 300 200 200 700
RbGb;EdlRtUvEklm¥KW
-Knøg (Path) EdleyIgeRbI³ + DB – DA + FA – FC + EC – EB D to B index = IDB = + $4 – $5 + $9 – $5 + $3 – $4 = + $2 ≥ 0
-KnøgEdleyIgeRbI³ + DC – DA + FA – FC D to C index = IDC = + $3 – $5 + $9 – $5 = + $2 ≥ 0
-KnøgEdleyIgeRbI³ + EA – FA + FC – EC E to A index = IEA = + $8 – $9 + $5 – $3 = + $1 ≥ 0
-KnøgEdleyIgeRbI³ + FB – FC + EC – EB
F to B index = IFB = + $7 – $5 + $3 – $4 = + $1≥ 0
edaysarkarKNnatamRbGb;EdlEklm¥ (Improvement Index) nImYy²mantémøviC¢man
b¤esµIsUnü dUcenHeKGacrkeXIjcemøIyGubTIm:al;ehIyEdl)anbgðajkñúgtaragsegçbxageRkam³
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
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36
taragTI10
Routes Units Shipped Per Unit Cost ($) Cost ($)
D to A E to B
E to C F to A
F to C
100 200
100 200
100
5 4
3 9
5
500 800
300 1800
500
Total Transportation Cost (TTC) 3900
]TahrN_TI4 Rkumh‘unmYymaneragcRk 4 kEnøgEdlplitplitpltambrimaNdUcxageRkam³
eragcRk brimaNplitpl
kMBt 70 T
PñMeBj 50 T
)at;dMbg 30 T
kMBg;qñaMg 20 T
Rkumh‘unenH RtUvkarCYlmeFüa)aydwkCBa¢ÚnplitpleTAkan;kEnøgtaMglk;EdlmanbrimaN
tRmUvkardUcxageRkam³
eKaledA brimaNtRmUvkar
RkugRBHsIhnu 60 T
PñMeBj 10 T
kMBg;cam 100 T
éføQñÜldwkCBa¢ÚnplitplBIedImTI (eragcRk) eTAkan;eKaledAnImYy²KitCa $/T bgðajkñúg
kaer:tUc²enARCugxagelI nigsþaMénRbGb;dUcxageRkam³
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37
taragTI1
edImTI
eKaledA RBHsIhnu PñMeBj kMBg;cam brimaNpÁt;pÁg;
kMBt 4 3 7 70
PñMeBj 5 2 10 50
)at;dMbg 13 8 17 30
kMBg;qñaMg 9 3 11 20
brimaNtRmUvkar 60 10 100 170
etIRkumh‘unRtUvcat;EcgkardwkCBa¢ÚnrebobNaedIm,I[R)ak;cMNayéfødwkCBa¢ÚnsrubTabbMput?
(1). rkcemøIydMbUgtamviFIeRCIserIsRbGb;RCugxageCIg-lic.
(2). rkcemøIydMbUgtamviFIeRCIserIsRbGb;manéføTabCageK.
(3). rkcemøIydMbUgtamviFIrktémøRbEhlv:UEhÁl. (4). rkéfødwkCBa¢ÚnGb,brmatamviFIsþibPIgsþÚn.
V- »nPaB (Degeneration)
kñúgkaredaHRsaycMeNaTkardwkCBa¢Únmþgmáal eKGacCYbnwg»nPaBEdlCakrNIminGac[
eKGacKNna ∆zij énRbGb;KµanTinñn½y Cij . »nPaBenHGacekIteLIgenAkñúgtaragcemøIydMbUg b¤
GacekIteLIgkñúgdMeNIrkarénviFIsþibPIgsþÚn.
kñúgdMeNaHRsayebICYbbBaðaenH eKRtÚvbMeBjkñúgRbGb;KµanTinñn½yEdlmantémødwkCBa¢Ún
TabCageKnUvGkSr E (Empty) nigcat;Tukfa E enHCaTinñn½ymYydUcTinñn½yd¾éTeTotEdr ehIyeRbI
viFIsþibPIgsþÚndUcFmµta. b:uEnþebIenAEtminGacKNna ∆zij énRbGb;KµanTinñn½y Cij )aneTot enaHeK
RtÚvbþÚrTItaMg E eTARbGb;EdlmantémødwkCBa¢ÚnTabbnÞab;…. ]TahrN_TI5 rkcemøIydMbUgtamviFIeRCIserIsRbGb;RCugxageCIg-lic rYcrkcemøIyRbesIrbMputtam
viFIsþibPIgsþÚnéncMeNaTkardwkCBa¢ÚnenAkñúg]TahrN_TI4Edl)anbgðajTinñn½ytamtaragTI1xag
eRkam.
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
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38
taragTI1
edImTI
eKaledA RBHsIhnu PñMeBj kMBg;cam brimaNpÁt;pÁg;
kMBt 4 3 7 70
PñMeBj 5 2 10 50
)at;dMbg 13 8 17 30
kMBg;qñaMg 9 3 11 20
brimaNtRmUvkar 60 10 100 170
VI- cemøIyRbesIrbMputmaneRcIn n igbERmbRmÜléføed ImÉkta
(Multiple Optimal Solution and Varying The Unit Cost)
♣ ♣♣ ♣♣
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39
lMhat ;
cUredaHRsaycMeNaTénkardwkCBa¢Ún 1 – 6 xageRkam³
k. rkcemøIydMbUgtamviFIeRCIserIsRbGb;RCugxageCIg-lic.
x. rkcemøIydMbUgtamviFIeRCIserIsRbGb;manéføTabCageK. K. rkéfødwkCBa¢ÚnGb,brmatamviFIsþibPIgsþÚn.
1. Rkumh‘unmYymaneragcRk 4 kEnøgEdlplitplitpltambrimaNnigvaRtÚvkarCYlmeFüa)ay
dwkCBa¢ÚnplitpleTAkan;kEnøgtaMglk; 3 kEnøgEdlmanbrimaNtRmUvkar. éføQñÜldwkCBa¢Únplit
plBIedImTI(eragcRk) eTAkan;eKaledAnImYy²KitCa $/T bgðajenARCugxagelInigxageqVgénRbGb;
dUcxageRkam³
eKaledA
edImTI RBHsIuhnu PñMeBj kMBg;cam
brimaN
plitpl
kMBt 4 3 7 70 t
PñMeBj 5 2 10 50 t
)at;dMbg 13 8 17 30 t
kMBg;qñaMg 9 3 11 20 t
brimaNtRmUvkar 60 t 10 t 100 t 170 t
etIRkumh‘unRtÚvcat;EcgkardwkCBa¢ÚnrebobNaedIm,I[R)ak;cMNayéføQñÜldwkCBa¢ÚnsrubTabbMput?
2. bBaðadwkCBa¢ÚnmYy[Tinñn½ydUctaragxageRkam³
Sink Sources
Department 1 ( $ / t )
Department 2 ( $ / t )
Department 3 ( $ / t )
Supplies ( t )
Factory 1 67 42
51 250
Factory 2 61 24
39 400
Factory 3 29 47
60 300
Factory 4 43 31
42 200
Demands ( t ) 400 150 600 1150
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
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40
rkéfødwkCBa¢ÚnGb,brma.
3. bBaðadwkCBa¢ÚnmYy[Tinñn½ydUctaragxageRkam³
Sink Sources
Department 1 ( $ / t )
Department 2 ( $ / t )
Department 3 ( $ / t )
Department 4 ( $ / t )
Supplies ( t )
Factory 1 1 5 3
4 100
Factory 2 4 2 2
4 60
Factory 3 3 1 2
4 120
Demands ( t ) 70 50 100 60 280
rkéfødwkCBa¢ÚnGb,brma.
4. bBaðadwkCBa¢ÚnmYy[Tinñn½ydUctaragxageRkam³
Sink Sources
Department 1 ( $ / t )
Department 2 ( $ / t )
Department 3 ( $ / t )
Department 4 ( $ / t )
Supplies ( t )
Factory 1 5 3
6 2 193
Factory 2 4 7
9 1 374
Factory 3 3 4
7 5 343
Demands ( t ) 163 182 307 258 910
rkéfødwkCBa¢ÚnGb,brma.
5. bBaðadwkCBa¢ÚnmYy[Tinñn½ydUctaragxageRkam³
Sink Sources
Department 1 ( $ / t )
Department 2 ( $ / t )
Department 3 ( $ / t )
Department 4 ( $ / t )
Department 5 ( $ / t )
Supplies ( t )
Factory 1 9 3
6 7 3 100
Factory 2 7 5
2 10 6 160
Factory 3 5 4
9 8 10 140
Demands (t) 90 60 80 100 70 400
rkéfødwkCBa¢ÚnGb,brma.
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
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41
6. Rkumh‘un Saussy Lumber dwkCBa¢ÚnTMnij pine flooring eTAkan;XøaMgpÁt;pÁg;eRKÓgsMNg;bIBI
tMbn; Pinevill, Oak Ridge nig Maple town . cUrkMNt;nUvEbbbTénkardwkCBa¢Ún[)anRbesIr
bMputcMeBaHTinñn½yEdlpþl;[enAkñúgtaragxageRkam.
Sink
Sources
Supply House 1
( $ / t )
Supply House 2
( $ / t )
Supply House 3
( $ / t )
Supplies
( t )
Pinevill 3 3
2 25
Oak Ridge 4 2
3 40
Maple town 3 2
3 30 Supply House Demand ( t ) 30 30 35 95
♣ ♣♣ ♣♣
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
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42
CMBUkT I5 RkahVmanTisedA n igbNþaj (Directed Graphs and Networks)
IV- esckþIepþIm (Introduction)
Baküfa RkahV manGtßn½yBIrepSgKñakñúgKNitviTüa. eyIgFøab;CYbrYcmkehIynUv
sBaØaNénRkahVtag[smIkar y = f(x). RkahVenH CarUbPaBmYytag[RKb;KU (x, y) EdlepÞóg
pÞat;smIkar. kñúgCMBUkenH eyIgsikSaGMBIRbePTmYyeToténRkahV. eBlenH eyIgminKitBIragExS
ekag (Knøg) EdlP¢ab;cMNucnanaeT b:uEnþfaetIvaman b¤KµanKnøgmanTisedA (pøÚvmanTisedA) mYy
rvagBIrcMNucenaH.
dUecñH RkahV KWCasMNuMéncMNuc b¤Fatu ehAfa fñaMg b¤kMBUl nigsMNuMénFñÚ (Knøg) EdlP¢ab;KU
énfñaMgTaMgenaH. RbsinebIFñÚTaMgenaHmanlMdab; EdlkarerobtamlMdab;enHbBa¢ak;BITisedAénFñÚ eK
ehARkahVenaHfa RkahVmanTisedAmYy.
eyIgnwgtagRkahVmanTisedAeday [N, L] Edl N CasMNuMénfñaMg ehIy L CasMNuMénFñÚ
manlMdab;EdlP¢ab;fñaMgenAkñúg N.
]TahrN_TI1 KUsRkahVmanTisedA [N, L] mYyEdl N = {1, 2, 3, 4} nig L = {(1, 2), (1, 3),
(1, 4), (2, 3), (3, 2), (3, 4)}.
kMNt;smÁal; rgVg; Loops minGacmaneTkñúgRkahVmanTisedAdUcCa (3, 3) KWCaFñÚminRtwmRtÚv.
m:aRTIsTMnak;TMng (Connection Matrix) énRkahVmYy KWCam:aRTIs M manFatu mij Edl
mij esµI 1 ebImanFñÚP¢ab;rvagfñaMg i nigfñaMg j b¤esµI 0 ebIKµanFñÚP¢ab;rvagfñaMg i nigfñaMg j mann½yfa
ijM m = Edl ij1 (i, j) L
m0 (i, j) L
∈= ∉
ebI
ebI .
KnøgmYyrvagfñaMg i nigfñaMg j énRkahV CasMNuMmanlMdab;énFñÚEdlP¢ab;fñaMg i nigfñaMg j Edl
fñaMgcugénFñÚnImYy²enAkñúgKnøgdUcKñanwgfñaMgedIménFñÚbnÞab; (minGacykFñÚCargVg;)aneT) ehIyKnøg
nImYy²pþl;nUvpøÚvmYysRmab;eFVIdMeNIr b¤Tak;TgrvagfñaMg i nigfñaMg j.
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43
]TahrN_TI2 rkm:aRTIsTMnak;TMngénRkahVmanTisedAkñúg]TahrN_TI1. rYcrkKnøgrvagfñaMg 1
nigfñaMg 4.
KYrkt;smÁal;fa RkahVmYy CaRkahVCab; b¤RkahVkuNik (Connected Graph) ebIRKb;KUén
fñaMgkñúgRkahVenaHmanKnøgmYyP¢ab;fñaMgTaMgenaH.
]TahrN_TI3 cUrKUsRkahVmYycMnYnEdlCaRkahVCab; b¤minEmnCaRkahVCab;.
kñúgCMBUkenH eyIgsikSaRkahVCab;manTisedA EdlsmtßPaBénFñÚmanTisedAP¢ab;fñaMgTaMg
BIr. RkahVTaMgenH ehAfa bNþaj (Networks).
bNþaj CaRkahVCab;manTisedAEdlcMnYnviC¢manRtUv)aneKkMNt;cMeBaHKnøgnImYy²
Edlvatag[dUcCacm¶ayrvagTIRkugBIr smtßPaBénEpñkrbs;bMBg;eRbgcMNayRbcaMqñaMénkarCYs
CulrvagqñaMmYy nigqñaMmYy. l.
II- karR)aRs½yTak;Tg (Communications)
kareRbIR)as;RkahVR)aRs½yTak;TgmYy KWedIm,Icg¥úlbgðajfaetIBaküccamGarammYyvaral
dalkñúgRkumy:agem:c. eyIgtagmnusSkñúgRkumedaycMNucénRkahV ehIytagpøÚvTak;TgKñaEdl
GacmanrvagmnusSTaMgenaHedayFñÚ. eyIgeRbIRkahVmanTisedA edIm,IbgðajfaFñÚmYyP¢ab;rvag
mnusS i eTAmnusS j mincaM)ac;naM[manFñÚP¢ab;rvagmnusS j nigmnusS i eT .
]TahrN_TI4 eK[RkahVmanTisedAénkarR)aRs½yTak;TgKñaxageRkam EdlFñÚTaMgenaHbgðajnUvkar
Tak;TgKñarvagmnusS 6 nak;. kMNt;m:aRTIsTMnak;TMngcMeBaHRkahVmanTisedAenH rYcehIyeFVIpl
bUkFatukñúgCYrQr nigCYredk nigTajrkB½t’manxøHBIlkçN³.
III- karRbkYtRbECg (Tournaments)
RkahVmanTisedAmYy CakarRbkYtRbECgmYy ebIvamanlkçN³mYyEdlcMeBaHRKb;KUén
cMNucepSgKña nig FñÚ b¤ sßitenAkñúgRkahVenaHEtminTaMgBIreT. eKk¾GacehARkahVénkarRbkYtRbECg
mYYyfaCaRkahVmanTisedAmanPaBlb;Cag BIeRBaHvak¾tag[TMnak;TMngrvagmnusSBIrnak;Edl
mnusSmñak;man\T§iBlelImñak;eTot.
BIRkahVmanTisedAEdleK[ eyIgkMNt;)anm:aRTIsTMnak;TMngmYy. RbsinebIeyIgbUkFatu
CMnaj ³ KNitviTüa karRsavRCavRbtibtþikar2
__________________________________________________________________________________Mr. Yim Ayuvathanak Vichea Operations Research II
44
tamCYredk eyIgTTYl)ancMnYnEdlbBa¢ak;BIkarQñHrbs;mnusSmñak;².
]TahrN_TI5 ]bmafa mnusS 4 nak;RbkYtKñaénkILa Tennis mYyEdlmanlkçN³CakarRbkYtCaCuM
(Round-robin Competition). lT§plmankñúgRkahVmanTisedAdUcxageRkam. kMNt;m:aRTIs
TMnak;TMngénRkahVenH rYcehIyeFVIplbUkFatukñúgCYredk nigTajrkB½t’manxøHBIlkçN³.
IV- cMeNaTGb,brmaénbNþaj (Network Minimization Problems)
enAkñúgcMeNaTGb,brmaénbNþaj b¤cMeNaTénedImeQIEdlduHEbkxøIbMput (Minimum
Spannig Tree) eyIgP¢ab;RKb;cMNucénbNþajedayKnøgEdlplbUkRbEvgrbs;Knøg (b¤EmkénedIm
eQI) TaMgenHmantémøGb,brma. ]TahrN_dUcCa eyIgP¢ab;GagsþúkeRbg (b¤Twk) enAkñúgTIRkugmYy
edaybMBg;EdlmancMnYnGb,brma b¤pþl;esvaTUrTsSn_ExSkabeTAtMbn;GPivDÆn_fµImYy. CaTUeTA
eKalbMNgenH KWcg;P¢ab;RKb;cMNucénbNþajenAeBlEdleKmindwgGMBITisedArbs;Knøg. edIm,I[
TMnak;TMngenHmanRbsiT§PaB eyIgminRtUv[manKnøg (b¤rgVg;) enAkñúgRkuménKnøgEdlP¢ab;RKb;
cMNucenaHeT.
]TahrN_TI6 bNþajxageRkamenHtag[sßanIy_mYy cMNucTI1 rbs;Rkumh‘unTUrTsSn_ExSkab nig
tMbn;GPivDÆn_fµI 5 (cMNucTI 2 dl;cMNucTI 6) EdlRkumh‘uncg;pþl;esva[. cMnYnenAelIKnøgTaMgenH
Cacm¶ay (KitCaKILÚEm:Rt) rvagcMNucnImYy². KYrkt;smÁal;fa edaysarEtmanRsTab;fµenAeRkam
dI eKminGacP¢ab;cMNucTI 1 cMNucTI 2 nigcMNucTI 5 eTAcMNucTI 6 b¤cMNucTI 3 eTAcMNucTI 5)aneT.
eyIgcg;rksMNuMénKnøgEdlpþl;nUvcMnYn (RbEvg) ExSkabGb,brmaEdlRtUveRbIedIm,IP¢ab;
tMbn;GPivDÆn_TaMgR)aM nigsßanIy_TUrTsSn_enH.
6 5 2 18 2 12 3 16 1 10 8 10 20 14 4 6 6
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V- cMeNaTrkRbEvgpøÚv x øIbMput (Shortest Route Problems)
enAkñúgcMeNaTrkRbEvgpøÚvxøIbMput eyIgcg;rkpøÚv (b¤Knøg) Edlpþl;nUvcm¶ayGb,brmaEdl
caM)ac;edIm,ItP¢ab;cMNucmYy EdleKehAfa cMNucedIm (b¤RbPB) eTAcMNucmYyeTotehAfa eKal
edA. mindUccMeNaTrktémøGb,brmaénbNþajeT eBlenHeyIgsikSaGMBITisedAénKnøgEdlP¢ab;
cMNucTaMgenaH RbPBmYynwgmanKnøgEtmYyKt;Edllatsn§wgecjBIRbPBenH ehIyeKaledAnwg
manKnøgEtmYyKt;Edl)anbBa©b;Rtwmva. eyIgmin[manrgVg;enAkñúgbNþajeT.
]TahrN_TI6 eK[bNþajxageRkamenH Edltag[RbB½n§pøÚvEteTA (One-way Road System)enA
kúñgTIRkugmYy. mankEnøgRbvtþisaRsþcMnYn 7 enAkñúgTIRkugdUcbgðajedaycMNucénRkahV. kMNt;rk
pøÚvRbesIrbMputEdlGñkeTscrKYrEteFVIdMeNIr ebIeKsßitenARtg;kEnøgTI1 ehIycg;eTAkEnøgTI7[)an
qab;tamEtGaceFVIeTA)an. ]bmafa cracrtampøÚvnImYy²manlkçN³dUcKña.
VI- cMeNaTrklMhUrGtibrma
CMnaj ³ KNit