数据模型与决策运筹学课后习题和案例答案017s.docx
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数据模型与决策运筹学课后习题和案例答案017s
CDSupplementtoChapter17
MoreAbouttheSimplexMethod
ReviewQuestions
17s.1-1No.
17s.1-2Theadjacentcornerpointsthatarebetterthanthecurrentcornerpointarecandidatestobethenextone.
17s.1-3Thebestadjacentcornerpointcriteriaandbestrateofimprovementcriteria.
17s.1-4Thesimplexmethodstartsbyselectingsomecornerpointastheinitialcornerpoint.
17s.1-5Ifnoneoftheadjacentcornerpointsarebetter（asmeasuredbythevalueoftheobjectivefunction）thanthecurrentcornerpoint,thenthecurrentcornerpointisanoptimalsolution.
17s.1-6Choosethebestadjacentcornerpoint.
17s.1-7Choosethenextcornerpointbypickingtheadjacentcornerpointhasthelowestobjectivefunctionvalueratherthanhighestwhengettingstarted.Theremayonlybeoneadjacentcornerpointifthefeasibleregionisunbounded.
17s.2-1Itisanalagoustostandinginthemiddleofaroomandlookingtowardonecornerwheretwowallsandthefloormeet.
17s.2-2Therearethree（atmost）adjacentcornerpoints.
17s.2-3Yes.
17s.2-4Withthreedecisionvariables,theconstraintboundariesareplanes.
17s.2-5Asystemofnvariablesandnequationsmustbesolved.
17s.3-1Thenamederivesfromthefactthattheslackvariablefora≤constraintrepresentstheslack（gap）betweenthetwosidesoftheinequality.
17s.3-2Anonnegativeslackvariableimpliesthattheleft-handsideisnotlargerthantheright-handside.
17s.3-3FortheWyndorproblem,theslackvariablesrepresentunusedproductiontimesinthevariousplants.
17s.3-4Itismuchsimplerforanalgebraicproceduretodealwithequationsthanwithinequalities.
17s.3-5Anonbasicvariablehasavalueofzero.
17s.3-6Abasicfeasiblesolutionissimplyacornerpointthathasbeenaugmentedbyincludingthevaluesoftheslackvariables.
17s.3-7Asurplusvariablegivestheamountbywhichtheleft-handsideofa≥constraintexceedstheright-handside.
17s.4-1
（1）Determinetheenteringbasicvariable;
（2） determinetheleavingbasicvariable;（3） Solveforthenewbasicfeasiblesolution
17s.4-2Theenteringbasicvariableisthecurrentnonbasicvariablethatshouldbecomeabasicvariableforthenextbasicfeasiblesolution.Amongthenonbasicvariableswithanegativecoefficientinequation0,choosetheonewhosecoefficienthasthelargestabsolutevaluetobetheenteringbasicvariable.
17s.4-3Theleavingbasicvariableisthecurrentbasicvariablethatshouldbecomeanonbasicvariableforthenextbasicfeasiblesolution.Foreachequationthathasastrictlypositivecoefficient（neitherzeronornegative）fortheenteringbasicvariable,taketheratiooftheright-handsidetothiscoefficient.Identifytheequationthathastheminimumratio,andselectthebasicvariableinthisequationtobetheleavingbasicvariable.
17s.4-4Theinitializationstepsetsuptostarttheiterationsandfindstheinitialbasicfeasiblesolution.
17s.4-5Examinethecurrentequation0.Ifnoneofthenonbasicvariableshaveanegativecoefficient,thenthecurrentbasicfeasiblesolutionisoptimal.
17s.4-6
（1）Equation0doesnotcontainanybasicvariables;
（2）eachoftheotherequationscontainsexactlyonebasicvariable;（3）anequation’sonebasicvariablehasacoefficientof1;（4）anequation’sonebasicvariabledoesnotappearinnanyotherequation.
17s.4-7Thetabularformperformsexactlythesamestepsasthealgebraicform,butrecordstheinformationmorecompactly.
Problems
17s.1
GettingStarted:
Select（0,0）astheinitialcornerpoint.
CheckingforOptimality:
Both（0,3）and（3,0）havebetterobjectivefunctionvalues（Z=6and9,respectively）,so（0,0）isnotoptimal.
MovingOn:
（3,0）isthebestadjacentcornerpoint,somoveto（3,0）.
CheckingforOptimality:
（2,2）hasabetterobjectivefunctionvalue（Z=10）,so（3,0）isnotoptimal.
MovingOn:
Movefrom（3,0）to（2,2）.
CheckingforOptimality:
（0,3）haslowerobjectivefunctionvalues（Z=6）,so（2,2）isoptimal.
17s.2
GettingStarted:
Select（0,0）astheinitialcornerpoint.
CheckingforOptimality:
Both（0,2.667）and（4,0）havebetterobjectivefunctionvalues（Z=5.333and4,respectively）,so（0,0）isnotoptimal.
MovingOn:
（0,2.667）isthebestadjacentcornerpoint,somoveto（0,2.667）.
CheckingforOptimality:
（2,2）hasabetterobjectivefunctionvalue（Z=6）,so（0,2.667）isnotoptimal.
MovingOn:
Movefrom（0,2.667）to（2,2）.
CheckingforOptimality:
（4,0）hasalowerobjectivefunctionvalues（Z=4）,so（2,2）isoptimal.
17s.3
GettingStarted:
Select（0,0）astheinitialcornerpoint.
CheckingforOptimality:
Both（0,5）and（4,0）havebetterobjectivefunctionvalues（Z=10and12,respectively）,so（0,0）isnotoptimal.
MovingOn:
（4,0）isthebestadjacentcornerpoint,somoveto（4,0）.
CheckingforOptimality:
（4,2）hasabetterobjectivefunctionvalue（Z=16）,so（4,0）isnotoptimal.
MovingOn:
Movefrom（4,0）to（4,2）.
CheckingforOptimality:
（3,4）hasabetterobjectivefunctionvalue（Z=17）,so（4,2）isnotoptimal.
MovingOn:
Movefrom（4,2）to（3,4）.
CheckingforOptimality:
（0,5）hasalowerobjectivefunctionvalues（Z=10）,so（3,4）isoptimal.
17s.4a）
GettingStarted:
Select（0,0）astheinitialcornerpoint.
CheckingforOptimality:
Both（2,0）and（0,5）havebetterobjectivefunctionvalues（Z=4and5,respectively）,so（0,0）isnotoptimal.
MovingOn:
（0,5）isthebestadjacentcornerpoint,somoveto（0,5）.
CheckingforOptimality:
（2,5）hasabetterobjectivefunctionvalue（Z=9）,so（0,5）isnotoptimal.
MovingOn:
Movefrom（0,5）to（2,5）.
CheckingforOptimality:
（2,0）hasalowerobjectivefunctionvalues（Z=4）,so（2,5）isoptimal.
b）GettingStarted:
Select（0,0）astheinitialcornerpoint.
CheckingforOptimality:
Movingtowardeither（2,0）or（0,5）improvestheobjectivefunctionvalue,so（0,0）isnotoptimal.
MovingOn:
Movingtoward（2,0）improvestheobjectivefunctionfasterthanmovingtoward（0,5）（arateof2vs.arateof1）,somoveto（2,0）.
CheckingforOptimality:
Movingtoward（2,5）improvestheobjectivefunctionvalue,so（2,0）isnotoptimal.
MovingOn:
Movefrom（2,0）to（2,5）.
CheckingforOptimality:
Movingtoward（0,5）lowerstheobjectivefunctionvalues,so（2,5）isoptimal.
17s.5a）
b）Theeightcornerpointsare（x1,x2,x3）=（0,0,0）,（0,0,1）,（0,1,0）,（0,1,1）,（1,0,0）,（1,0,1）,（1,1,0）,and（1,1,1）.
c）ObjectiveFunction:
Profit=x1+2x2+3x3
OptimalSolution:
（x1,x2,x3）=（1,1,1）andProfit=6.
CornerPoint（x1,x2,x3）
Profit=x1+2x2+3x3
（0,0,0）
0
（0,0,1）
3
（0,1,0）
2
（0,1,1）
5
（1,0,0）
1
（1,0,1）
4
（1,1,0）
3
（1,1,1）
6
d）Thesimplexmethodwouldstartat（0,0,0）,movetothebestadjacentcornerpointat（0,0,1）,thento（0,1,1）,andfinallytotheoptimalsolutionat（1,1,1）.
17s.6a）
b）Thetencornerpointsare（x1,x2,x3）=（0,0,0）,（2,0,0）,（2,2,0）,（1,3,0）,（0,3,0）,（0,0,2）,（2,0,2）,（2,2,2）,（1,3,2）,（0,3,2）
c）ObjectiveFunction:
Profit=2x1+x2–x3
OptimalSolution:
（x1,x2,x3）=（2,2,0）andProfit=6.
CornerPoint（x1,x2,x3）
Profit=2x1+x2–x3
（0,0,0）
0
（2,0,0）
4
（2,2,0）
6
（1,3,0）
5
（0,3,0）
3
（0,0,2）
–2
（2,0,2）
2
（2,2,2）
4
（1,3,2）
3
（0,3,2）
1
d）Thesimplexmethodwouldstartat（0,0,0）,movetothebestadjacentcornerpointat（2,0,0）,andthentotheoptimalsolutionat（2,2,0）.
17s.7a）s1=10–x2
s2=20–2x1–x2
b）s1≥ 0ands2≥0.
c）x2+s1=10
2x1+x2+s2=20
d）Valuesoftheslackvariablesat（x1,x2）=（10,0）ares1=10ands2=0.
Theequationsfortheconstraintboundarylinesonwhich（10,0）liesare
x2=0
2x1+x2=20
Thecorrespondingbasicfeasiblesolutionis（x1,x2,s1,s2）=（10,0,10,0）.
Thebasicvariablesarex1ands1;thenonbasicvariablesarex2ands2.
17s.8a）25x1+40x2+50x3≤ 500.
b）s≥0.
c）s=0.
17s.9a）ObjectiveFunction:
Profit=2x1+x2
OptimalSolution:
（x1,x2）=（4,3）andProfit=11
CornerPoint（x1,x2）
Profit=2x1+x2
（0,0）
0
（5,0）
10
（4,3）
11
（0,5）
5
b）Thegraphicalsimplexmethodwouldstartat（0,0）,movetothebestadjacentcornerpointat（5,0）,andfinallymovetotheoptimalsolutionat（4,3）.
c）3x1+2x2+s1=15
x1+2x2+s2=10
d）
BasicFeasibleSolution（x1,x2,s1,s2）
BasicVariables
NonbasicVariables
（0,0,15,10）
s1,s2
x1,x2
（5,0,0,5）
x1,s2
x2,s1
（4,3,0,0）
x1,x2
s1,s2
（0,5,5,0）
x2,s1
x1,s2
e）Thegraphicalsimplexmethodwouldstartat（0,0,15,10）,movetothebestadjacentcornerpointat（5,0,0,5）,andfinallymovetotheoptimalsolutionat（4,3,0,0）.
17s.10a）s1=2x1+3x2–21.
s2=5x1+3x2–30.
b）s1≥ 0ands2≥0.
c）2x1+3x2–s1=21.
5x1+3x2–s2=30.
d）Valuesofthesurplusvariablesat（x1,x2）=（3,5）ares1=0ands2=0.
Theequationsfortheconstraintboundarylinesonwhich（3,5）liesare
2x1+3x2=21
5x1+3x2=30
Thecorrespondingbasicfeasiblesolutionis（x1,x2,s1,s2）=（3,5,0,0）.
Thebasicvariablesarex1andx2;thenonbasicvariablesares1ands2.
17s.11a）20x1+10x2≥100.
b）s≥0.
c）s=0.
17s.12a）
GettingStarted:
Select（16,0）astheinitialcornerpoint.（Cost=32.）
CheckingforOptimality:
Both（15,0）and（0,24）havebetterobjectivefunctionvalues（Cost=30and24,respectively）,so（16,0）isnotoptimal.
MovingOn:
（0,24）isthebestadjacentcornerpoint,somoveto（0,24）.（Cost=24.）
CheckingforOptimality:
（0,20）hasabetterobjectivefunctionvalue（Cost=20）,so（0, 24）isnotoptimal.
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