Lab Session No. 01
b b b
Objective:
IntroductionbtobOperationsbResearch
Theory:
ThebOriginsbofbOperationsbResearch:
ThebrootsbofbORbcanbbebtracedbbackbmanybdecades,bwhenbearlybattemptsbwerebmadebtobuseba
bscientificbapproachbinbthebmanagementbofborganizations.bHowever,bthebbeginningbofbthebactivitybca
lledboperationsbresearchbhasbgenerallybbeenbattributedbtobthebmilitarybservicesbearlybinbWorldbWarbI
I.bThebBritishbandbthenbthebU.S.bmilitarybmanagementbcalledbuponbablargebnumberbofbscientistsbtobap
plybabscientificbapproachbtobdealingbwithbthisbandbotherbstrategicbandbtacticalbproblems.bInbeffect,bth
eybwerebaskedbtobdobresearchbonb(military)boperations.bThesebteamsbofbscientistsbwerebthebfirstbORb
teams.b
Whenbthebwarbended,bthebsuccessbofbORbinbthebwarbspurredbinterestbinbapplyingbitboutsidebth
ebmilitarybasbwell.bBybthebearlyb1950s,bthesebindividualsbhadbintroducedbthebusebofbORbtobabvarietybo
fborganizationsbinbbusiness,bindustry,bandbgovernment.b
IntroductionbtobOperationsbResearch:
Operationsbresearchbisbanbinterdisciplinarybbranchbofbappliedbmathematicsbthatbusesbmethod
sbsuchbasbmathematicalbmodeling,bstatistics,bandbalgorithmsbtobarrivebatboptimalborbnearboptimalbsol
utionsbtobcomplexbproblems.bItbisbtypicallybconcernedbwithbdeterminingbthebmaximab(ofbprofit,basse
mblyblinebperformance,bcropbyield,bbandwidthbetc)borbminimab(ofbloss,briskbetc.)bofbsomebobjectivebf
unction.bOperationsbresearchbhelpsbmanagementbachievebitsbgoalsbusingbscientificbmethods.
ThebtermbOperationsbResearchb(OR)bdescribesbthebdisciplinebthatbisbfocusedbonbthebapplicati
onbofbinformationbtechnologybforbinformedbdecision-
making.bInbotherbwords,bORbrepresentsbthebstudybofboptimalbresourceballocation.bThebgoalbofbORbisb
tobprovidebrationalbbasesbforbdecisionbmakingbbybseekingbtobunderstandbandbstructurebcomplexbsitua
tions,bandbtobutilizebthisbunderstandingbtobpredictbsystembbehaviorbandbimprovebsystembperformanc
e.b
TORAbisbabprominentbsoftwarebpackagebforblinearbprogrammingbandbitsbextensions.bItbisbsimpleband
bhasbanbeasybtobusebinterfacebforbfirst btimebusers.b
MuhammadbAbubakarbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb2016-IM-109
,SolvingbLPbProblemsbinbTORA:
ClickbLinearbProgrammingbatbthebmainbscreen.
NowbselectbGObTObINPUTbSCREENbtobstartbabproblem.bClickbSelectbExistingbFilebifbyoubhavebalre
adybsavedbthebdatabandbwantbtobreopenbit.
b
NowbenterbthebdatabandbthenbclickbonbSOLVEbMenu.bAlsobsavebyourbdata.
MuhammadbAbubakarbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb2016-IM-109
,Abnewbscreenbappearsbaskingbyoubthebtypebofbsolutionbyoubwant.bSelectbAlgebraicbandbthenbFinalbS
olutin.
Torabwillbshowbthebresultbforbthisbmodel.bInbthisbwaybwebcanbsolveblinearbProgrammingbmathematic
albmodelsbonbthisbsoftware.
MuhammadbAbubakarbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb2016-IM-109
, Lab Session No. 02
b b b
Objective:
IntroductionbtobLINEARbPROGRAMMING
Theory:
IntroductionbtobLinearbProgramming:
Thebdevelopmentbofblinearbprogrammingbhasbbeenbrankedbamongbthebmostbimportantbscient
ificbadvancesbofbthebmid-
20thbcentury.bTodaybitbisbabstandardbtoolbthatbhasbsavedbmanybthousandsborbmillionsbofbdollarsbforbm
ostbcompaniesborbbusinessesbofbevenbmoderatebsizebinbthebvariousbindustrializedbcountriesbofbthebwo
rld;bandbitsbusebinbotherbsectorsbofbsocietybhasbbeenbspreadingbrapidly.bThebmostbcommonbtypebofbap
plicationbinvolvesbthebgeneralbproblembofballocatingblimitedbresourcesbamongbcompetingbactivitiesb
inbabbestbpossibleb(i.e.,boptimal)bway.bMorebprecisely,bthisbproblembinvolvesbselectingbtheblevelbofbc
ertainbactivitiesbthatbcompetebforbscarcebresourcesbthatbarebnecessarybtobperformbthosebactivities.bTh
ebchoicebofbactivityblevelsbthenbdictatesbhowbmuchbofbeachbresourcebwillbbebconsumedbbybeachbactiv
ity.bThebvarietybofbsituationsbtobwhichbthisbdescriptionbappliesbisbdiverse,bindeed,brangingbfrombtheba
llocationbofbproductionbfacilitiesbtobproducts,btobtheballocationbofbnationalbresourcesbtobdomesticbne
eds,bfrombportfoliobselectionbtobthebselectionbofbshippingbpatterns,bfrombagriculturalbplanningbtobthe
bdesignbofbradiationbtherapy,bandbsobon.
BasicbConceptsbofbLinearbProgramming
ObjectivebFunction:b
ThebObjectivebFunctionbisbablinearbfunctionbofbvariablesbwhichbisbtobbeboptimizedbi.e.,bmaxi
mizedborbminimized.be.g.,bprofitbfunction,bcostbfunctionbetc.b
Constraints:b
Ablinearbequationbrepresentsbabstraightbline.bLimitedbtime,blaborbetc.bmaybbebexpressedbasblin
earbinbequationsbandbarebcalledbconstraints.
Optimization:b
Abdecisionbwhichbisbconsideredbthebbestbone,btakingbintobconsiderationballbthebcircumstance
sbisbcalledbanboptimalbdecision.bThebprocessbofbgettingbthebbestbpossibleboutcomebisbcalledboptimizat
ion.
SolutionbofbLP:b
Absetbofbvaluesbofbthebvariablesbx1,bx2,….,xnbwhichbsatisfyballbthebconstraintsbisbcalledbthebsol
utionbofbthebLP.
FeasiblebSolution:b
Absetbofbvaluesbofbthebvariablesbx1,bx2,bx3,….,xnbwhichbsatisfyballbthebconstraintsbandbalsobthe
bnon-negativitybconditionsbisbcalledbthebfeasiblebsolutionbofbthebLP.
OptimalbSolution:b
Thebfeasiblebsolution,bwhichboptimizesb(i.e.,bmaximizesborbminimizesbasbthebcasebmaybbe)bt
hebobjectivebfunctionbisbcalledbtheboptimalbsolution.b
MathematicalbFormulationbofbLinearbProgrammingbProblems:
Therebarebfourbstepsbinbthebmathematicalbformulationbofblinearbprogrammingbproblembasbabmath
ematicalbmodel.bWebwillbdiscussbformulationbofbthosebproblemsbwhichbinvolvebonlybtwobvariables.b
1. Identifybthebdecisionbvariablesbandbassignbsymbolsbxbandbybtobthem.bThesebdecisionbvariable
sbarebthosebquantitiesbwhosebvaluesbwebwishbtobdetermine.
MuhammadbAbubakarbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb2016-IM-109
b b b
Objective:
IntroductionbtobOperationsbResearch
Theory:
ThebOriginsbofbOperationsbResearch:
ThebrootsbofbORbcanbbebtracedbbackbmanybdecades,bwhenbearlybattemptsbwerebmadebtobuseba
bscientificbapproachbinbthebmanagementbofborganizations.bHowever,bthebbeginningbofbthebactivitybca
lledboperationsbresearchbhasbgenerallybbeenbattributedbtobthebmilitarybservicesbearlybinbWorldbWarbI
I.bThebBritishbandbthenbthebU.S.bmilitarybmanagementbcalledbuponbablargebnumberbofbscientistsbtobap
plybabscientificbapproachbtobdealingbwithbthisbandbotherbstrategicbandbtacticalbproblems.bInbeffect,bth
eybwerebaskedbtobdobresearchbonb(military)boperations.bThesebteamsbofbscientistsbwerebthebfirstbORb
teams.b
Whenbthebwarbended,bthebsuccessbofbORbinbthebwarbspurredbinterestbinbapplyingbitboutsidebth
ebmilitarybasbwell.bBybthebearlyb1950s,bthesebindividualsbhadbintroducedbthebusebofbORbtobabvarietybo
fborganizationsbinbbusiness,bindustry,bandbgovernment.b
IntroductionbtobOperationsbResearch:
Operationsbresearchbisbanbinterdisciplinarybbranchbofbappliedbmathematicsbthatbusesbmethod
sbsuchbasbmathematicalbmodeling,bstatistics,bandbalgorithmsbtobarrivebatboptimalborbnearboptimalbsol
utionsbtobcomplexbproblems.bItbisbtypicallybconcernedbwithbdeterminingbthebmaximab(ofbprofit,basse
mblyblinebperformance,bcropbyield,bbandwidthbetc)borbminimab(ofbloss,briskbetc.)bofbsomebobjectivebf
unction.bOperationsbresearchbhelpsbmanagementbachievebitsbgoalsbusingbscientificbmethods.
ThebtermbOperationsbResearchb(OR)bdescribesbthebdisciplinebthatbisbfocusedbonbthebapplicati
onbofbinformationbtechnologybforbinformedbdecision-
making.bInbotherbwords,bORbrepresentsbthebstudybofboptimalbresourceballocation.bThebgoalbofbORbisb
tobprovidebrationalbbasesbforbdecisionbmakingbbybseekingbtobunderstandbandbstructurebcomplexbsitua
tions,bandbtobutilizebthisbunderstandingbtobpredictbsystembbehaviorbandbimprovebsystembperformanc
e.b
TORAbisbabprominentbsoftwarebpackagebforblinearbprogrammingbandbitsbextensions.bItbisbsimpleband
bhasbanbeasybtobusebinterfacebforbfirst btimebusers.b
MuhammadbAbubakarbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb2016-IM-109
,SolvingbLPbProblemsbinbTORA:
ClickbLinearbProgrammingbatbthebmainbscreen.
NowbselectbGObTObINPUTbSCREENbtobstartbabproblem.bClickbSelectbExistingbFilebifbyoubhavebalre
adybsavedbthebdatabandbwantbtobreopenbit.
b
NowbenterbthebdatabandbthenbclickbonbSOLVEbMenu.bAlsobsavebyourbdata.
MuhammadbAbubakarbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb2016-IM-109
,Abnewbscreenbappearsbaskingbyoubthebtypebofbsolutionbyoubwant.bSelectbAlgebraicbandbthenbFinalbS
olutin.
Torabwillbshowbthebresultbforbthisbmodel.bInbthisbwaybwebcanbsolveblinearbProgrammingbmathematic
albmodelsbonbthisbsoftware.
MuhammadbAbubakarbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb2016-IM-109
, Lab Session No. 02
b b b
Objective:
IntroductionbtobLINEARbPROGRAMMING
Theory:
IntroductionbtobLinearbProgramming:
Thebdevelopmentbofblinearbprogrammingbhasbbeenbrankedbamongbthebmostbimportantbscient
ificbadvancesbofbthebmid-
20thbcentury.bTodaybitbisbabstandardbtoolbthatbhasbsavedbmanybthousandsborbmillionsbofbdollarsbforbm
ostbcompaniesborbbusinessesbofbevenbmoderatebsizebinbthebvariousbindustrializedbcountriesbofbthebwo
rld;bandbitsbusebinbotherbsectorsbofbsocietybhasbbeenbspreadingbrapidly.bThebmostbcommonbtypebofbap
plicationbinvolvesbthebgeneralbproblembofballocatingblimitedbresourcesbamongbcompetingbactivitiesb
inbabbestbpossibleb(i.e.,boptimal)bway.bMorebprecisely,bthisbproblembinvolvesbselectingbtheblevelbofbc
ertainbactivitiesbthatbcompetebforbscarcebresourcesbthatbarebnecessarybtobperformbthosebactivities.bTh
ebchoicebofbactivityblevelsbthenbdictatesbhowbmuchbofbeachbresourcebwillbbebconsumedbbybeachbactiv
ity.bThebvarietybofbsituationsbtobwhichbthisbdescriptionbappliesbisbdiverse,bindeed,brangingbfrombtheba
llocationbofbproductionbfacilitiesbtobproducts,btobtheballocationbofbnationalbresourcesbtobdomesticbne
eds,bfrombportfoliobselectionbtobthebselectionbofbshippingbpatterns,bfrombagriculturalbplanningbtobthe
bdesignbofbradiationbtherapy,bandbsobon.
BasicbConceptsbofbLinearbProgramming
ObjectivebFunction:b
ThebObjectivebFunctionbisbablinearbfunctionbofbvariablesbwhichbisbtobbeboptimizedbi.e.,bmaxi
mizedborbminimized.be.g.,bprofitbfunction,bcostbfunctionbetc.b
Constraints:b
Ablinearbequationbrepresentsbabstraightbline.bLimitedbtime,blaborbetc.bmaybbebexpressedbasblin
earbinbequationsbandbarebcalledbconstraints.
Optimization:b
Abdecisionbwhichbisbconsideredbthebbestbone,btakingbintobconsiderationballbthebcircumstance
sbisbcalledbanboptimalbdecision.bThebprocessbofbgettingbthebbestbpossibleboutcomebisbcalledboptimizat
ion.
SolutionbofbLP:b
Absetbofbvaluesbofbthebvariablesbx1,bx2,….,xnbwhichbsatisfyballbthebconstraintsbisbcalledbthebsol
utionbofbthebLP.
FeasiblebSolution:b
Absetbofbvaluesbofbthebvariablesbx1,bx2,bx3,….,xnbwhichbsatisfyballbthebconstraintsbandbalsobthe
bnon-negativitybconditionsbisbcalledbthebfeasiblebsolutionbofbthebLP.
OptimalbSolution:b
Thebfeasiblebsolution,bwhichboptimizesb(i.e.,bmaximizesborbminimizesbasbthebcasebmaybbe)bt
hebobjectivebfunctionbisbcalledbtheboptimalbsolution.b
MathematicalbFormulationbofbLinearbProgrammingbProblems:
Therebarebfourbstepsbinbthebmathematicalbformulationbofblinearbprogrammingbproblembasbabmath
ematicalbmodel.bWebwillbdiscussbformulationbofbthosebproblemsbwhichbinvolvebonlybtwobvariables.b
1. Identifybthebdecisionbvariablesbandbassignbsymbolsbxbandbybtobthem.bThesebdecisionbvariable
sbarebthosebquantitiesbwhosebvaluesbwebwishbtobdetermine.
MuhammadbAbubakarbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb2016-IM-109
