karRsavRCavRbtibtþikar2 · 2020. 6. 13. · CnM aj ³ KNitviTüa karRsavRCavRbtibtþikar2 _____ Mr. Yim Ayuvathanak Vichea Operations Research II

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



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



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



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 .



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.

♣ ♣♣ ♣♣



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



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.



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?

♥



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.



10

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;



11

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 )



12

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



13

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.


Maximize profit z = 3x1 + 4x2

Subject to 0.3x1 + 0.5x2 ≤ 300

x1 + 1.5x2 ≤ 750

x1, x2 ≥ 0

dMeNaHRsaytamviFIsIumepøc[taragcemøIycugeRkay³



14

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 .


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



15

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



16

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.


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.



17


x. begáItcMeNaTeTVPaBnigedaHRsayvatamviFIsIumepøc rYcTajrkcemøIyéncMeNaTTaMgBIr.

6. eK[cMeNaTdMbUgdUcxageRkam³ Maximize profit P = 80 x1 + 75 x2 Subject to x1 + 3x2 ≤ 4 2x1 + 5x2 ≤ 8 x1, x2 ≥ 0


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



18

x1 + 3x2 − 2x3 ≥ 20 and x1, x2, x3 ≥ 0.



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.



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.

♥



19

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).



20

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

♣ ♣♣♣♣



21

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)+ ≤



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



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?

♥



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.



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



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



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



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



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;



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


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



31

taragTI4


Des Moines ( D ) $5

$4 $3 100

100

Evansville ( E ) $8 $4 200

$3 100

300

Fort Lauderdale ( F ) $9 300

$7 $5 300


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



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;



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


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


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



34

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 )


Des Moines ( D ) $5 100 $4

$3

100

Evansville ( E )

$8 100

-

$4 200

+

$3 300


$9 100 +

$7

-

$5 200

300



-KnøgEdleyIgeRbI³ + DB – DA + EA – EB D to B index = IDB = + $4 – $5 + $8 – $4 = + $3



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)


Des Moines ( D ) $5 100 $4

$3

100

Evansville ( E )

$8 –

$4 200

Start $3 + 100

300


$9 200 +

$7

$5 – 100

300



-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



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



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.



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)

♣ ♣♣ ♣♣



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 )



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



40

rkéfødwkCBa¢ÚnGb,brma.


Sink Sources





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



Sink Sources





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



Sink Sources






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




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

♣ ♣♣ ♣♣



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.



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



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



45

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

Documents

karRsavRCavRbtibtþikar2 · 2020. 6. 13. · CnM aj ³ KNitviTüa karRsavRCavRbtibtþikar2 _____ Mr. Yim Ayuvathanak Vichea Operations Research II