数据、模型与决策(运筹学)课后习题和案例答案009s.doc
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数据、模型与决策(运筹学)课后习题和案例答案009s.doc
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CDSUPPLEMENTTOCHAPTER9
SOMEPERSPECTIVESONSOLVING
BINARYINTEGERPROGRAMMINGPROBLEMS
ReviewQuestions
9s-1 Pushtheobjectivefunctionlineinthedirectionofimprovingvaluesoftheobjectivefunction.Stopatthelastinstantwhentheobjectivefunctionlinepassesthroughafeasibleintegerpoint(abinarysolution).
9s-2 Theexhaustiveenumerationmethodcanbeusedforlargerproblemswhilethegraphicalmethodislimitedtoproblemswithjusttwovariables.
9s-3 Theexhaustiveenumerationmethodquicklybecomesunwieldyifthenumberofvariableisincreasedverymuch.
9s-4 Forproblemswithmorethanafewvariables,itisgenerallyeasiertosolvelinearprogrammingproblemsthanBIPproblemsofthesamesize.
9s-5 TheLPrelaxationofaBIPproblemreplacestheconstraintoneachbinaryvariablethatthevariableisbinarybytheconstraintthatitisbetween0and1.
9s-6 TheLPrelaxationisrelevantforhelpingtosolveaBIPproblembecausethesolutionmayendupbeingthesolutionfortheBIPproblem,andifnotitatleastgivesagoodplacetobeginthesearchforanoptimalsolution.
9s-7 ABIPproblemcontainingmutuallyexclusivealternativeisanexampleofaproblemwithspecialstructure.
9s-8 ThetwoprimarydeterminantsofcomputationaldifficultyforaBIPproblemarethenumberofbinaryvariablesandanyspecialstructureintheproblem.
9s-9 OnemajorpitfallwiththeroundingprocedureisthatitmayproduceasolutionthatisinfeasiblefortheBIPproblem.Theotherpitfallisthatthereisnoguaranteethatafeasiblesolutionwillbeoptimal,orevennearlyoptimal,fortheBIPproblem.
Problems
9s-1 a)
Solution
Feasible?
P=2x1+5x2
Optimal?
(0,0)
Yes
0
(1,0)
No
(0,1)
Yes
5
***
(1,1)
No
b) Optimalsolution:
(x1,x2)=(0,1).
c) SolvingtheLPrelaxationgraphically,theoptimalsolutionis(x1,x2)=(1,0.667).Thisroundsto(1,1)whichisnotafeasiblesolution.
d) Roundingdownresultsinasolutionof(1,0)whichisalsonotafeasiblesolution.
9s-2 a)
Solution
Feasible?
P=–5x1+25x2
Optimal?
(0,0)
Yes
0
(1,0)
Yes
–5
(0,1)
No
(1,1)
Yes
20
***
b) Optimalsolution:
(x1,x2)=(1,1).
c) SolvingtheLPrelaxationgraphically,theoptimalsolutionis(x1,x2)=(0,0.9).Thisroundsto(0,1)whichisnotafeasiblesolution.
d) Roundingdownresultsinasolutionof(0,0)whichisafeasiblesolutionbutyieldsZ=0whichinnotanoptimalsolution.
9s-3
Solution
Feasible?
P=9x1+5x2+6x3+4x4
Optimal?
(0,0,0,0)
Yes
0
(1,0,0,0)
Yes
9
(0,1,0,0)
Yes
5
(0,0,1,0)
No
(0,0,0,1)
No
(1,1,0,0)
Yes
14
***
(0,1,1,0)
No
(0,0,1,1)
No
(1,0,0,1)
No
(1,0,1,0)
No
(0,1,0,1)
Yes
9
(1,1,1,0)
No
(0,1,1,1)
No
(1,0,1,1)
No
(1,1,0,1)
No
(1,1,1,1)
No
9s-4 a) True.ThecurrentalgorithmsforsolvingBIPproblemsstillarenotnearlyasefficientasthoseforsolvinglinearprogrammingproblems.
b) True.ThetwoprimarydeterminantsofcomputationaldifficultyforaBIPproblemare
(1)thenumberofbinaryvariablesand
(2)anyspecialstructureintheproblem.Thissituationisincontrasttolinearprogramming,wherethenumberof(functional)constraintsismuchmoreimportantthatthenumberofvariables.
c) False.OnemajorpitfallwiththeroundingprocedureisthatitmayproduceasolutionthatininfeasiblefortheBIPproblem.
CD9s-4
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