lecture05

lecture05 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 1 TheSimplexMethodII YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/˜yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 2 TheTransportationProblem min ∑ mi =1 ∑ n j =1 c ij x ij s.t. ∑ n j =1 x ij = s i , ∀ i ∑ mi =1 x ij = d j , ∀ j x ij ≥ , ∀ i,j. Assumethatthetotalsupplyequalthetotaldemand.Thus,exactlyoneequality constraintisredundant. Ateachstepthesimplexmethodattemptstosendunitsalongaroutethatis unused(non-basic) inthecurrentBFS,whileeliminatingoneoftheroutesthatis currentlybeing used(basic) . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 3 1 2 3 n 1 2 m . . . . . . s1 s2 sm Supply Demand d1 d2 d3 dn C11, x11 Figure1:Supplychainnetwork Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 4 SupplyChainNetwork Retailers Warehouses 1 2 3 4 Supply 1 12 13 4 6 500 2 6 4 10 11 700 3 10 9 12 4 800 Demand 400 900 200500 20000 Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 5 TransportationSimplexMethod:PhaseI 1.Startwiththecellinthe northwestcornercell 2.Allocateasmanyunitsaspossible,consistentwiththe available supplyand demand. 3.Moveonecellto right ifthereisremainingsupply;otherwise,moveonecell down . 4.gotoStep2. 500 700 800 400 900 200 500 Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 6 ConstructtheinitialBFS 400 100 700 800 900 200 500 Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 7 ConstructtheinitialBFS 400 100 700 800 800 200 500 Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 8 ConstructtheinitialBFS 400 100 700 800 100 200 500 Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 9 ConstructtheinitialBFS 400 100 700 100 700 200 500 Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 10 ConstructtheinitialBFS 400 100 700 100 200 500 500 Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 11 ConstructtheinitialBFS 400 100 700 100 200 500 Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 12 1 2 1 2 s1 s2 d1 d2 d3 C11, x11 d4 4 3 s3 Demand Supply 3 Figure2:ABFShasatreenetworkstructure Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 13 TransportationSimplexMethod:PhaseII 1.Determinethe shadowprices foreachsupplyside u i andeachdemandside v j from u i + v j = c ij forevery USED cell(basicvariable)(whichcorrespondtostepsolving A T B y = c B ).Onecanalwaysset v n =0 byviewingthelastdemand constraintredundant. 2.Calculatethe reducedcosts r ij = c ij- u i- v j forthe UNUSED cells(non-basicvariable)(whichcorrespondtostep computing r N = c N- A T N y ).Ifthereducedcostforeveryunusedcellis nonnegative,thenSTOP:declare OPTIMAL . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05 14 3.Selectanunusedcellwiththe mostnegative reducedcost.Usinga chain reactioncycle ,determinethe maximumnumber ofunits( θ )thatcanbe allocatedtotheenteringcellandadjusttheallocationappropriately.Update thevaluesofthe newsetofUSED cells(BFS). 4.GotoStep1. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #05...
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lecture05 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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