Bus Routes in Destin Florida - Other Transportation in Destin
Table of Contents€¦ · The Transportation Method of linear programming is applied to the...
13
Table of Contents • Abstract ................................................................................. 2 • Introduction .......................................................................... 2 • Define the problem ............................................................... 5 • Results and Discussion (problem solving) ............................. 6 • Conclusion ........................................................................... 13
Transcript of Table of Contents€¦ · The Transportation Method of linear programming is applied to the...
Table of Contents
• Abstract.................................................................................2
• Introduction..........................................................................2
• Definetheproblem...............................................................5
• ResultsandDiscussion(problemsolving).............................6
• Conclusion...........................................................................13
• Abstract
Worldwide,themanufacturingindustryisnowaimingforcompetitiveadvantageandincreased
profits.Inthispaper,wegiveanoverviewofstate-of–theartoperationsresearchmodeland
techniquesusedintransportingthealuminumfromthesupplierstothedemandedindustries.
Wedescribetheplanningproblemsarisingtherediscusssomemodelsandalgorithmstosolve
them.
• Introduction
Aluminumisthethirdmostabundantelementinnature–comprisingsomeeightpercentofthe
earth’scrust.Todaymorealuminumisproducedeachyearthanallothernon-ferrousmetals
combined.Aluminumislight,strong,conductive,durable,flexibleandeasytorecycle.Amonga
widediversityofapplicationsfromartandcraftstohightechnology,thethreemainfieldsin
whichaluminumisusedaretransportation,packagingandbuildingandelectricalmachine
tools.Whilediscussingthealuminumindustrythefirstpointthatcomestoourmindisthe
productionandtheshipmentofaluminum.
Theprimaryaluminumindustryisoneoftheoldestandmostimportantmanufacturing
industriesthatGCCcountrieshavereliedupon,todeveloptheirindustrialsectorsand
economies.
ThealuminumindustrystartedwiththeestablishmentofAluminumBahrain(ALBA),in1968,
followedbytheestablishmentofDubaiAluminumCompanyLimited"Dubal"in1975.Inrecent
yearstheinterestinaluminumindustrywidelyincreased,andtheGulfStatesbecameina
positiontobeamongthelargestproducersofaluminumintheworld,followingthe
establishmentofEmiratesAluminum(EMAL)inAbuDhabi.Inaddition,AbuDhabimaintains
greatpotentialforgrowth,consistentwiththedevelopmentagendaoftheEmirate,the
"EconomicVision2030"viewsmetalindustries,includingaluminum,astheenginesoffuture
economicgrowth,whichwouldspurachievingtheobjectivesoftheemirateregarding
economicdiversification.
Themainobjectiveofthisreportistodefinetheproblemfacedinthealuminumindustries
whichisincreasingtheprofitoftransportingthealuminumamongdifferentcustomersby
reducingtheshipmentcostbyusingoperationresearchmethodssuchastransportationmodel.
Operationsresearchhasplayedanimportantroleintheanalysisofproblemsolvingand
decisionmakingthatisusefulinthemanagementoforganization.Inoperationsresearch,
problemsarebrokendownintobasiccomponentsandthensolvedindefinedstepsby
mathematicalanalysis.
Theprocesscanbebroadlybrokendownintothreesteps:
1. Asetofpotentialsolutionstoaproblemisdeveloped.
2. Thealternativesderivedinthefirststepareanalyzedandreducedtoasmallsetof
solutionsmostlikelytoproveworkable.
3. Thealternativesderivedinthesecondsteparesubjectedtosimulatedimplementationand,
ifpossible,testedoutinreal-worldsituations.
TheTransportationMethodoflinearprogrammingisappliedtotheproblemstostudythe
efficientoftransportationroutesi.e.howefficientlytheproductfromdifferentsourcesof
productionistransportedtothedifferentdestinations,suchasthetotaltransportationcostis
minimum.
Furthermore,forsolvingthetransportationproblem,thefollowingstepsaretobe
systematicallyfollowed:
1. Obtainingtheinitialfeasiblesolution,whichmeansidentifyingthesolutionthatsatisfies
therequirementsofdemandandsupply.Thereareseveralmethodsthroughwhichthe
initialfeasiblesolutioncanbeobtained;theseare:
o North-westcorner
o Leastcostmethod
o Vogel’sapproximationmethod
2. Testingtheoptimalityoftheinitialfeasiblesolution.Oncethefeasiblesolutionis
obtained,thenextstepistocheckwhetheritisoptimumornot.Therearetwomethods
usedfortestingtheoptimality:
o Stepping-stoneMethod
o ModifiedDistributionMethod(MODI)
3. Thefinalstepistorevisethesolutionuntiltheoptimumsolutionisobtained
• Define the problem TheAluminumCompanyhadthreebranchesamongthecountry.EmalwhichislocatedinAbu
Dhabi,Al-Taweelaareaanditsupplies1.5milliontonofaluminumyearly.Secondly,wehave
DubalwhichislocatedinDubai,Jabil-Aliareaanditsupplies1milliontonyearly.Andfinally,
thereisAlbawhichislocatedinwesternregionofUAEanditsupplies971,420tonyearly.There
arefivebigindustriesinthecountrythatgettheirAluminumdemandfromthosethree
suppliers.Firstly,Aerospaceindustrythatrequire159,642tonsofAluminumyearly.Secondly,
Electronicsindustrywhichrequires140,214tonsofAluminumyearly.Thirdly,Construction
industrythatisthelargestamongthemintermofAluminumdemandsinceitrequires
1,687,139tonsyearly.Fourthly,cansandpackagesindustrythatrequiredademandof415,141
tonsyearly.Finally,Automotiveindustrythatrequireademandof1,069,284tonsyearly.The
costoftransportingonetonofAluminumyearlytothededicatedindustryisasfollow:
Aerospace C&P Automotive Construction Electronics
Emal28,550,325
AED/year
114,201,300
AED/year
399,704,550
AED/year
571,006,500
AED/year
28,550,325
AED/year
Dubal19,033,550
AED/year
76,134,200
AED/year
266,468,700
AED/year
380,671,000
AED/year
19,033,550
AED/year
Alba73,958,285
AED/year
125,729,084
AED/year
147,916,569
AED/year
332,812,281
AED/year
59,166,628
AED/year
Tosimplifythecalculation,thecostwasroundedintonearestmillionasshowninthe
belowtable(TransportcostfromSuppliertoindustry(millionAED/ton)):
Aerospace C&P Automotive Construction Electronics
Emal 29AED/year 114AED/year 400AED/year 571AED/year 29AED/year
Dubal 19AED/year 76AED/year 266AED/year 381AED/year 19AED/year
Alba 74AED/year 126AED/year 148AED/year 333AED/year 59AED/year
• Results and Discussion (problem solving) Inordertominimizethetransportationcostbetweenthesupplierandthereceiving
industryatransportationmodelwasused.
! Thedecisionvariableisdefinedtobe:
• j=thesupplier(Emal,Dubal,Alba)
• i=theindustry(Aerospace,C&P,Electronics,Construction,Automotive)
• !!" =tonsofAluminumbeingtransportedfromsuppliertoindustry
! TheObjectivefunctionisusedtominimizethetransportationcost:
• Z=29!!" + 114!!"# + 400!!"# + 571!!" + 29!!! + 19!!" + 76!!"# +
266!!"# + 381!!" + 19!!" + 74!!! + 126!!"# + 148!!!" + 333!!" +
59!!"
! Theconstrainsareillustratedasfollowing:
• !!" + !!"# + !!"# + !!" + !!! = 1,500,000
• !!" + !!"# + !!"# + !!" + !!" = 1,000,000
• !!! + !!"# + !!!" + !!" + !!" = 971,420
• !!" + !!" + !!! = 159,642
• !!"# + !!"# + !!"# = 415,141
• !!"# + !!"# + !!!" = 1,069,284
• !!" + !!" + !!" = 1,687,139
• !!! +!!" + !!" = 140,214
• !!" ≥ 0
! Constructingthetransportationtableau:
A CP AU C E Supply
E 1,500,000
D 1,000,000
A 971,420
Demand 159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
29 114 400 571 29
193812667619
74 126 148 333 59
! InstructthestartingbasicfeasiblesolutionusingNorthWestCornerMethod:
A CP AU C E Supply
E
159,642 415,141 925,217
1,500,000
1,340,358
925,217
0
D
144,067 855,933
1,000,000
855,933
0
A
831,206 140,214
971,420
140,214
0
Demand159,642
0
415,141
0
1,069,284
144,067
0
1,687,139
831,206
0
140,214
03,471,420
Transportationcost,Z=29!!" + 114!!"# + 400!!"# + 571!!" + 29!!! + 19!!" +
76!!"# + 266!!"# + 381!!" + 19!!" + 74!!! + 126!!"# + 148!!!" + 333!!" + 59!!"
! Z=29 ∗ 159642+ 114 ∗ 415141+ 400 ∗ 925217+ 266 ∗ 144067+ 381 ∗
855933+ 333 ∗ 831206+ 59 ∗ 140214 = 1,071,539,011 !"#/!"#$%!
29 114 400 571 29
193812667619
74 126 148 333 59
! InstructthestartingbasicfeasiblesolutionusingMinimumCellCostMethod:
A CP AU C E Supply
E x x x
1,500,000
x
1,500,000
0
D
159,642 415,141 97,864 187,139 140,214
1,000,000
840,358
700,144
285003
187,139
0
A x x
971,420
x x971,420
0
Demand159,642
0
415,141
0
1,069,284
97,864
0
1,687,139
1,500,000
0
140,214
0
3,471,420
Transportationcost,Z=29!!" + 114!!"# + 400!!"# + 571!!" + 29!!! + 19!!" +
76!!"# + 266!!"# + 381!!" + 19!!" + 74!!! + 126!!"# + 148!!!" + 333!!" + 59!!"
! Z=571 ∗ 1500000+ 19 ∗ 159642+ 76 ∗ 415141+ 266 ∗ 97864+ 381 ∗ 187139+
19 ∗ 140214+ 148 ∗ 971420 = 1,134,849,923 !"#/!"#$%!
29 114 400 571 29
193812667619
74 126 148 333 59
! InstructthestartingbasicfeasiblesolutionusingVogel’sApproximationMethod(VAM):
A CP AU C E Supply
E 159,642 415,141 97,864
687,139
140,214
1,500,000
0
00
D
x x x 1,000,000 x
1,000,000
0
00
A x x
971,420
x x
971,420
0
15
Demand159,642
0
415,141
0
1,069,284
97,864
0
1,687,139
687,139
0
140,214
0
3,471,420
10
10
38
38
118
134
48
190
10
10
Transportationcost,Z=29!!" + 114!!"# + 400!!"# + 571!!" + 29!!! + 19!!" +
76!!"# + 266!!"# + 381!!" + 19!!" + 74!!! + 126!!"# + 148!!!" + 333!!" + 59!!"
! Z=29 ∗ 159642+ 114 ∗ 415141+ 400 ∗ 97864+ 571 ∗ 687139+ 29 ∗ 140214+
381 ∗ 1000000+ 148 ∗ 971420 = 973,246,681 !"#/!"#$%!
29 114 400 571 29
193812667619
74 126 148 333 59
Onceaninitialsolutionisderived,thetransportationcostproblemmustbesolvedusing
eitherthestepping-stonemethodorthemodifieddistributionmethod(MODI).Inthisproject
MODImethodwasusedtoobtaintheoptimalsolutionfromabasicfeasiblesolution.MODIisa
modifiedversionofthestepping-stonemethodinwhichmathequationsreplacethestepping-
stonepaths.TheresultfoundusingVAMmethodwillbeasaninitialsolutiontotheMODI
method.
!! = !!" = !!" = !! = !! = Supply
A CP AU C E
!! = E
159,642
415,141
97,864
687,139
140,214
1,500,00
!! =D
X
x
x
1,000,000
x
1,000,000
!! =A
x
x
971,420
x
x
971,420
Demand
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
29
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
114
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
400
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
571
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
29
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
19
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
19
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
76
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
266
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
381
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
74
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
126
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
148
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
333
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
59
159,642 415,141 1,069,284 1,687,139 140,214 3,471,420
! Computedforallcellswithallocations:!! + !! = !!" =unittransportationcostfor
!"##!".
o !!" = !! + !! = 29
o !!"# = !! + !!" = 114
o !!"# = !! + !!" = 400
o !!" = !! + !! = 571
o !!! = !! + !! = 29
o !!" = !! + !! = 381
o !!!" = !! + !!" = 148
Sevenequationswitheightunknowns,thereforelet!! =0andsolvetoobtain:
o !! = 29, !!" = 114, !!" = 400, !! = 571, !! = 29
o !! = −190, !! = −252
! Usefollowingtoevaluateallemptycells:!!" − !! − !! = !!" ,wherekijequalsthecost
increaseordecreasethatwouldoccurbyallocatingtoacell.Fortheemptycellsinthe
previoustable:
o !!": !!" = !!" − !! − !! = 19-29+190=180
o !!"#: !!"# = !!"# − !!" − !! =76–114+190=152
o !!"#: !!"# = !!"# − !!" − !! = 266− 400+ 190 = 56
o !!": !!" = !!" − !! − !! = 19− 29+ 190 = 180
o !!": !!" = !!" − !! − !! = 74− 29+ 252 = 297
o !!"#: !!"# = !!"# − !!" − !! = 126− 114+ 252 = 264
o !!": !!" = !!" − !! − !! = 333− 571+ 252 = 15
o !!!: !!! = !!! − !! − !! = 59− 29+ 252 = 282
Sincenonegativeanswerappearedinthecalculationabove,thismeanthatthetableobtained
usingtheVAMmethodistheoptimalsolution,andtheminimumtransportationcostcanbe
achievedis 973,246,681 !"#/!"#$%!.
• Conclusion
Inconclusiontheoperationresearchandanalyticsdriveperformanceandchangein
organizationsofalltypes–largeandsmall,privateandpublic,for-profitandnot-for-profit.
Usingtechniquessuchasmathematicalmodelingtoanalyzecomplexsituations,O.R.and
Analyticsenablemoreeffectivedecisionsandmoreproductivesystemsbasedonrobustdata,
thefullerconsiderationofavailableoptions,andcarefulpredictionsofoutcomesandestimates
ofrisk.Inourcase,transportingthealuminuminUAEfromvarioussupplierstodifferent
destination,thesuppliersare(EMAL,DUBAL&Alba)&theindustrialthatneedsthealuminum
are(aerospace,electronics,construction,automotive,cansandpackage).Themainaimofour
casewastominimizetheshipmentcostbyusingoneoftheoperationresearchmathematical
methodwhichistransportationmodelanditincludesthreemethodswhichare(North-west
corner,leastcostmethodandvogel’sapproximationmethod).Moreover,bydemonstratedthe
threemethodsthelowershipmentcostwas973,246,681 !"#/!"#$%!byusingtheVAM
method,afterthatwetesttheoptimalsolutionbyusingmodifieddistributionmethod(MODI),
theresultfoundusingVAMmethodwillbeasaninitialsolutiontotheMODImethod.TheVAM
methodistheoptimalsolution,andtheminimumtransportationcostcanbeachievedis
973,246,681 !"#/!"#$%!.
