Solution Manual for Operations Research: An Introduction, 10th Edition

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SolutionsmanualOperationsResearch:AnIntroductionTenthEditionHamdyA.TahaUniversityofArkansas/Fayetteville

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DownloadedfromStudyXY.com®+StudyXYSdYe.o>\|iF’prE\3SStudyAnythingThisContentHasbeenPostedOnStudyXY.comassupplementarylearningmaterial.StudyXYdoesnotendroseanyuniversity,collegeorpublisher.Allmaterialspostedareundertheliabilityofthecontributors.wv8)www.studyxy.com

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2wvContents1WhatisOperationsResearch?2ModelingwithLinearProgramming3TheSimplexMethodandSensitivityAnalysis4DualityandPost-OptimalAnalysis5TransportationModelanditsVariants6NetworkModels7AdvancedLinearProgramming8GoalProgramming9IntegerLinerProgramming10HeuristicProgramming11TravelingSalespersonProblem(TSP)12DeterministicDynamicProgramming13InventoryModeling(withIntroductiontoSupplyChains)14ReviewofProbability15DecisionAnalysisandGames16ProbabilisticInventoryModels17QueuingSystems18SimulationModeling19MarkovChains20ClassicalOptimizationTheory21NonlinearProgrammingAlgorithmsAppendixC~~AMPLmodelingLanguagei

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2wvChapter1WhatisOperationsResearch?1-1

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>©5aChapter1iWeeks1-4:2weekendtripsFYV-DEN-FYVand2weekendtripsDEN-FYV-DEN.Week|(a)LetT=totaltimetomoveallfour5:1regulartrip.individualstotheothersideoftheriver.TheCost:4x320+1x400=$2200objectiveistodeterminethetransferschedule|thatminimizesT.2(b)Lett=crossingtimefromonesidetothe.A|other.Usecodes1,2,5,and10torepresent(a)GivenafenceoflengthL:ClNyJi()h=3L,w=2L,Area=06L2Amy,Jim,John,andKelly.(2)h=IL,w=4L,Area=_04L_Solution(2)isbetterbecausetheareaislargerERIa—(L=2(w+h),w=1/2-h(500(a=)[2z=wh=h(L/2-h)=Lh/2-h?IL[G.10)—(t=10[25,10|52/8h=L172-2h=0[2[=20@[5.10|Thos,h=L/4andw=L/4.[none[(1,2)~¢=2,[12510|Solutionisoptimalbecausezisaconcavefunction||Total=2+1+10+2+2=17minutesx=cumulativenumberofdropsofballs#1and#2Satanyfloor(problemvnknown)yi=floorfromwhichithdropofball#1occurs.JimStep0:Setyo=0,y;=x,andi=1.CurveFastGeneralstepi:Dropball#]fromflooryi.IfitisJCurve|.500.200dented,useball#2tocheckfloorsyi+1toy,-1,°°Fast|.100.300inthatorder.Else,if#1isnotdented,seti=i+1,andrepeatstepi.—Formulafordeterminingy;:Broaeeoffastball,y:mustincludethe(cumulative)i#1-dropsfromJoe:PrepareforcurveorfastballoorifloorYieuwotthesamenumberol(b)JoetriestoimprovehisbattingscoreandJimThpsatanyTooryi,pscannotexceedX=1.|rie10counterJoe'sactionbyselectingaless—Fxeit])favorablestrategy.ThismeansthatneitherplayerEkeSEGSPNwillbesatisfiedwithasingle(pure)strategy.cx|folx2Doi(=3Theproblemisnotanoptimizationsituationinreaswi)=ix-(i-1)thefamiliarsenseinwhichtheobjectiveisMaximumnumberof#1-dropsisx(elseYiYi|7imizedorminimized.Instead,theconflictingfori>fehighestfloorfromwhich#1situationrequiresacompromisesolutioninwhichEl_222neitherplayeristemptedtochangestrategy.GameYo=X=(e102=X"x12+x/2=(xi02theory(Chapter14)providessuchasolution.Fora100-storeybuilding,y,>100,orx*+x—200>0.Theassociatedquadraticequationyieldsx=13.64and~14.64.Theroundedpositivevaluex=14isthesmallestintegerthatsatisfiestheinequality.1-2

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>Chapter13a\LLLetL=ops.1and2=20sec,C=ops.3and4=25sec,U=op.5=20sec.Gantchart:L1=loadhorse1,L2=loadhorse2,etc.onejoist:0---L1-—20-—C1-—45-—U1+L1-—85——U2+L2-—125—~-U1+L1—165—U2+H.2---20520-12-4045-—-C2--—-7085—-C1--110125—-C2---140165-C1-190205---C2--230--U2--250Total=250Loadersutilization=[250-(5+25))/250=88%Cutterutilization=[250-(20+15+15+15+15)]/250=68%twojoists:0-—2L1---40—2C1—90—-2(U1+L1)—170——2C1---220—2U1---26040—-2L2—8090—2C2-—140170--2U2--210Total=260Loadersutilization=[260-(10+10)]/260=92%Cutterutilization=[260-(40+30+40)]/250=58%threejoists:0-—3L1--60—-3C1——-135—---—-3C2-—-—-~210—-3U2--—27060---3L.2-—120135-—-3U1-—-195Total=270Loadersutilization=[270-(15+15))/270=89%Cutterutilization=[270-(60+60)]/270=56%Recommendation:Onejoistattimegivesthesmallesttime.Theproblemhasotheralternativesthatcombine1,2,and3joists.Cutterutilizationindicatesthatcutterrepresentsthebottleneck.Notethatall‘dotsareindistinguishableeveniftheyaredesignatedas1,2,3,...,10.(a)Alternative1:Movedots5,6,and7belowbottomrow,movedots8and9belownew5,6,and7.Move10tothebottom.Numberofmoves=6.Alternative2:Seepart(b).(b)Threemoves:Movedot1uptotheleftofdot8,dot4totherightofdot9,anddot10belowdots2and3.108956712341-3

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>Chapter1=8wn@Alternative1:Breakoneendlinkofeachchainandconnecttoanotherchain.Fourbreaksandre-solders,cost=4x(2+3)=20cents.Alternative2:SeePart(b)(b)Breakthreelinksinonechainandusethemtoconnecttheremainingthreechains:Threebreaksandre-solder,cost=3x(2+3)=15cents.9Representtheselected2-digitnumberas10x+y.Thecorrespondingsquarenumberis10x+y-(x+y)=9x.Thismeansthattheselectedsquarewillalwaysbe9,18,27,...,or81.Byassigningzerodollarstothesesquares,therewardisalwayszeroregardlessoftherewardsassignedtotheremainingsquaresorthenumberoftimesthegameisrepeated.Assignasequentialnumberxtoeachcartons,xeX={1,2,3,4,5,6,7,8,9,10}.(a)LetYbethesctofcartonssofarweighed(initiallyY=@).GeneralStep:RandomlyselectacartonyeX-Y.Ifyweighs90oz,stop.Else,augmentytoYandrepeattheGeneralStep.1<numberoftimesscaleisused<10.(b)Exactlyonce!TakexbottlesfromcartonxeXtoendupwith(142+...+10)=(10+11)/2=55bottles.Weighthe55bottles.Iftheweight=550~x,cartonxisthedefectiveonc.1-4

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TT]aCHAPTER2ModelingwithLinearProgramming|21

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Chapter2=3a@XX,21or=,+x,21[1]Quanlydideriontsreounllshe14](b)X+2x,23andx,+3x,%€Fllrergronbntenechsfsnchin:(Yxz2Xor=X,so@Fh23SK+4Ke,os%ETTre=FerSX5%20z=@)(5,40)=(14%)X72xX,%)=e2Ysxtax,Exl+yxy=22<2¢(xi+2x4=9gh&infeasibleTheodaalirrcammt-CoLiralia]®)(x,0)=(22)aoaAnisfergpiam.NondondondyxIe)zNCanbeaccoondidgh+4Hicaerbxztxy277pd4aigmixedanligpoipummin]1x2+2==yxr+1x2=o<Ifeasible(cloptes-|x2=z=zz=sSx2+9x2=$18(0)C5,%,)=(3,15)XX,20Ex3+4x1.8=24=2¢Vx3+2x)is=626}finerkls“1X3+1XLS==1.85ixeg=1.52ZzSx3+4x15=F221WBELxd=2,1)X),X,Z©ex+axi216<RYLpocole(x2+2x1=4<6~ixz+lxl==)<xy=<zZ=Sx2+4x1=$14©(ox)=(22-1)x,zo,x,<0,wferutleConclenni(c)goveothfetfesaihleSeldon(Xs%)=(2,2);de5,and5,beHeunetdddocly[3|Emandy&MianM2.Formis5,=24=(6x45)=4fms[dayFrMA1Sy==f,+2x)=€-(araxz)=0tonsJoy2-2

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.Chapter2)32X-©-X+x,=1fet%referencepaint:Yo[5]XaGplimuincrdxX(12)2=$/3-5sk¢*gx4ZX-.©(©X4%ap.©)6x,+94224Gotimiim.:6.2)2gFORGm2hd4%iPS.[6]©)NofeasibleSpace+%...8].So|fedrigpe=daywryGfpradackca)Cb)<242Matimuze2=2X,+3x,KA¥s.t.weSY10%,+Sx;=6oo[0]JPXe6X,+2X,s6&o0(2)&x,+10X,<=goo©td)‘=loo&oN)a]A@x%a-imum:9xpAtda)%rAd2=#/9-50Soxey8x,22fSSxapprmuen:°c\og!(2,2).r-tisOplirasrn.OCcursodA:4%=52.99Xp=(4-12x’2=%148.24continued.—2-3

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Chapter2oo3K=numberdfumilsffAEIBYTiXp=Tuber.ofsonalofBX=¢weetamBMaximizez=20x+50%MaximizeZ=-05X+-08X25.4ITPrmx,+57,$0X,2-25(44%)p+KzX,©-S(XHK)Xx,£looXx,=SX2x,+4X,©240XX&S000X,Xp20Xi,X20vo2)2Optimum50\3LsX,=180*x,2500ey,X=#2500B|vz=Fazsi7a7X=>~;ad):5;OptimaloccasafB-ee%=number.Apradiadernie2)X=80unitsAy=rams§AmariconanXy=RoUnitsMaireZ=/S00X,+1000%z$2,600sg+.Xx,+X,£30X,=ruarmberof51pectsfay.10]x2/0+Pullenatresve%>10aximize2=r‘2StXiKegKakZoFooGoo(Xeophmum:ox,S50,9SXE580KFX=20.Xo%,+%$30X=1000%2,£580#Zz="Yg,0fessex,2\0°°Cphimisen°3a(byChangex40,$30Ho4%,€3/°700ulGetz=#4500Optrmmrnbobuchaz=P41500-go,000=F150;Fologn:.wo%=€50cheessConclusion:Anyizedcrumresp©Xoo18743barswiltLepheprodiiad=2=$285%vo2-4

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>Chapter23=wvX=undzgtHluliosA3]apeX=ariloffArla.s120fagnoKEP7774Kui=xeaF=8X,+10x,aARE6SX,FSX,=/S0LEAEISNE%x,>Bodoo3oo,<152Ophraum:Xm200,X,=C0,72=$26150©=z40Aresallocation:673Grane,$3)whenXa»£200x,=play4forcl502ssiwlrorsy,18)MaxeruzeZ=2X+x,SE.xX,+x,Sl0ESERSPY)3Nphimdyladers'eEa%,=4Rrwes53Kya€ewes<42=14"pleasarits”-:NNa«FT10x,po\';helONaclx,DiymipGpfot16]peNXo.»Dubybr.4Byps-MaximizeZ=8%+5K2oeSb.AX+XxS900aX,</508X,€200AolHEEENNX,.5,30a"400ZtX=Nbr.ofFransLoxeo14)optimum;d.Xp=nbr.ofchestboxesX=f00Gpa!MagimizeZ=X*135%,200£200X,220Ger2S54.2K19%,Z€0Kee?z=51800x,<200XxX,€/20.X,,%,20200x,continued.continued.2-5

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>Chapter23aX)=AadipmiradaVT]OptimimoccursaA:Xo=TVrumelioX,=5128tonsporGurMagimizez=a5Xp=70.256FonsLomaimg2a>Z-53546eofSeem-ypack£5Ophmalrats=5028_5Xi>2zor-x+2X,$010-256xX;12(4)RIX,+.9X,2(zo+)=21X,£900,X,,%Z°Ghimumz=16[538thofSham200|XAZ=161538-153844=74921bty0x,=Nbr.ofradicommencints19.beyondthefrst7usaX,=Abe.§TVadsbyord7firstwo|MaximizeZ=2000+3000X,+Sort2000ky,S-t.Z00(X+1)+2000(,+1)<20,0005Xr300X;=loveo300CK+)©Fx200002000(XK,+1)€-§X20,0002.oC)<Page.Soo1000onOplempnnoceeensatA:Makrmize.Z=R000,+3000X,17900X=60.61winulss770®X=30-3wrinubisSF3ogx,+200%,=17700z=§/8.183c0x,&/S700®71400X=tonsofCGConsumedptr18]000%=°®Xe»FonsfCsamasmedpinRewr1sXp20Maximize7=/R000X,+9000X,Xsst.1800XX,+2/00),£2006(X%+X,)i~200X,+[0ax,$0®2X,+TX,=20B85Xy5X,202-¥2&EY208Na.xOptimumblohrm»wt&RadioCommevcesds=2.3340=53.33roTVads=(+1=2ZN-aPyP1814j=)5Z=107666.6777000=/I4Ey=ESI=Fo==3continued2-6

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>Chapter23-=7)X=rusnbondhlponKvn20]X=numb4)HiFiwneds22oxamb.9)blissgainfonex=PoemdenofHiFiwrilsMaxonizeZ=§X+126|Conolraunds:|se&.Ex+4X,©4F0x-9=F32Aox,+60KF25x60=/S00(1)SX,+SX,SAPOX-5E=412.8Tox,+60X,©3xé0s2100@)9%,+6X2<EFoX-8Y=422.42X,+4X,€5X=300(3)ov’XkzoEX,+4X2TS,=432sx,485%.+5,=HZ.§XpophmumatA:4x,+6Xy+Sg=422.4X=1(2Shietsfo..|xo=2Bousea[RnObjectivefunction:sl\@z=$348[bvMinimizeS,+5+353=RETI-1SK7Isx)QA|Thao,monS,+5,+5;=maxISHT/SX:7hMaximizeZ=ISX,*/SX,HeAx6x,+4x,<%320Xogx+5x%=+z.8@X,=Wb.ofdeskspenday=ax,6X,=922416)X,=Nbr.ofChairsfondag{©¥,3X,20Lo.6)..MaxmwmzeZ=50%+/00X,A(Atternaliveophimax51?Ko,Xeo<yo7,¥,=50.8unisErr\Xo231.63unitsX,%ogy®INGF21238.4min750ToxX,£/20,x,€60(3,4)TNoue4INaOphimumooxX;Id60[~yA~NPTrwNESg.Wit,x|CmremwnaA©.0Ophmum:B“02c|Xx=90skHpop2@pTIMUN)Xy=HYCharsE1.2)17

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Chapter23%Xo3“-[2830FEA,z*4Soox(+:,5|y“SoetodoElxOpliamismnX=o,x,=v,Z=o,“whichismonSorsizal(c}2=¥%,=ntumboroffersfosethinStore4HlMXg=AlientecefRevdfuseck+nStured7]llQdditirnalconstraint:x<45029MinimizeZ=&X76x2s.t.2Xx2205sXx</21[rr]“esx,£/020Optimum"Soe138xX,aq500000Co4fF26woNMGplismarndwn:.x=450IbOplomum:X=3501X=10Rewrs2.$450Xx;=70Krsurs)2=140Stessadexcontinued.continued...2-8

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Chapter25FT32]33EA=LifemToX=15aridonblclaphedLdXp=10bt)dary,form3Xa=104nicelid“Lehted,atkReferingcapacityZ=X+X,[0bid’KMining2=X+X,Miniorizez=X,+Xp-.Subgect~&Lutpict-&dX,task,2/0C4KX,!2aXz4CKTR)6X,—4h20—6x,FoX,<0x.xRX+lXK,=1d12KyZoASK,+6X,230To.IX,4sex,210GAaplimimsilodirro:ISK)£1,2FXsXe20phimIonsFormToes:LINEARPROGRAMMING—GRAPHICALOLUTIONFEivamSpiteLINEARPROGRAMING~GRAPHICALSOLUTIONFEELDeedstoobes~~OOOOOO-—iSamEt’Taemn?oo/OpeimiumwAN:ehTTREe2-9

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