8 Transportation Problem Alpha-Beta

Transcription

1 8 rnsportton Proble Alph-Bet Now, we ntroduce n ddtonl lgorth for the Htchcock rnsportton proble, whch ws lredy ntroduced before hs s the Alph-Bet Algorth It copletes the lst of soluton pproches for solvng ths well-known proble he Alph-Bet Algorth s prl-dul soluton lgorth Owng to the splcty of the dul proble, ths procedure s cpble of usng sgnfcnt nsghts nto the proble structure Busness Coputng nd Opertons Reserch 76

2 8. Proble defnton nd nlyss Refresh: he prl proble c : Delvery costs for ech product unt tht s trnsported fro suppler to custoer, : otl supply of,..., b : otl dend of,..., n x : Quntty tht suppler,..., delvers to the custoer,..., n, P Mnze c x n... n s.t x... n E b n En En En E n (,,, n,, n,, n ) x x,..., x,..., x,..., x,..., x..., x,..., x Busness Coputng nd Opertons Reserch 76

3 nd the correspondng dul n n D Mxze π + b π α + b β s.t. + n En c, n En c, n E n n E n α... E c n π,... E c n, β,... E n... En c n E n, n n E n c, n.e., { } { },...,n :,..., :α + β c, Busness Coputng nd Opertons Reserch 76

4 Drect Observton he dul consders soewht odfed proble hs y be nterpreted s follows here s thrd prty tht offers trnsportton servce between the plnts nd the consuers For ths servce, both sdes hve to py n ndvdul fee. Specfclly, the th suppler pys α nd the th consuer β Obvously, t s not possble to chrge ore thn c, for the respectve cobnton Otherwse, snce t possesses ore effcent lterntve, the copny would not ke use of ths lterntve hus, the dfference c, -α -β s denoted s specultve gn of the consdered copny Consequently, whenever ths dfference s negtve, the prl proble s hold to ntroduce (,) n the bss. Otherwse, we better keep t out. Busness Coputng nd Opertons Reserch 76

5 he frst row of the prl tbleu If we consder the frst row of the prl tbleu, we drectly obtn c, c, cb AB A c, π A c, A π c α β, If we hve c, <, the dul vrbles re not fesble nd outsourcng s not resonble. Busness Coputng nd Opertons Reserch 764

6 Fesble dul solutons n+ Obvously, snce c, we hve π s, trvl ntl soluton. hs trvl soluton cn be drectly proved by { }, β n c,..., { } α n c β,...,n, Busness Coputng nd Opertons Reserch 765

7 Consder n exple α ( 5 6) b ( 6 ) Genertng n ntl soluton : β ( ) n{,,, } n{,,, } n{ 4,5,6, } c n n n 4 5 {,,, } { } { },,,,,5, 6 Busness Coputng nd Opertons Reserch 766

8 Busness Coputng nd Opertons Reserch 767 Exple the soluton s obvously fesble hus, we get, Wth α α α α α β β β c β

9 Preprng the Prl-Dul Algorth In order to prepre the Prl-Dul Algorth, we ntroduce: { } IJ, α + β c. hus, we obtn the reduced prl (RP) Mnze ( ) ( n+ ) ( IJ ) ( IJ ), s.t., x E, A, IR, b IR x b x x Mnze, x IJ n n+, s.t.,,,, IJ +,,, IJ x { } x + x,,..., { } x + x b,,,..., n x x ( IJ ) Busness Coputng nd Opertons Reserch 768

10 Preprng the Prl-Dul Algorth Mnze x, s.t., n+ x + x,,...,,, IJ +,, IJ ( IJ ) { } x + x b,,..., n x x { } Busness Coputng nd Opertons Reserch 769

11 8. Anlyzng the reduced prl (RP) Obvously, t holds: n n x + x, b x + + x,, IJ, IJ Snce totl dend nd supply re dentcl, we hve n n b x + x, x + +, IJ, IJ n n x x, x + x,, IJ, IJ + + x, Busness Coputng nd Opertons Reserch 77

12 Anlyzng (RP) x +, x +, x ( ) IJ (, ) ( ) IJ (, ) ( ) IJ (, ) n x x x, Obvously,t holds :, + Hence, we conclude: n n x + + x, IJ, x,, IJ n n x + + x, IJ n x Busness Coputng nd Opertons Reserch 77

13 Drect concluson Altogether, we therefore obtn: x + x, x + x,, IJ, IJ x x,, IJ n n n n x + + x, x + + x, b, IJ, IJ n n + n n x b + x, x + b x,, IJ, IJ Busness Coputng nd Opertons Reserch 77

14 Consequences + n n x + b x,, IJ Snce nzng deternes the obectve functon of the reduced prl of the Htchcock rnsportton Proble, we ust hve to xze, IJ x, hs leds to the followng RP : Mxze x, s.t.,, IJ, { },,,, IJ, IJ { } x,, x,,..., x b,,..., n Busness Coputng nd Opertons Reserch 77

15 Anlyzng the proble n detl Dn! hs proble rends e of soethng Defntely! It s ust MAX-FLOW! PROBLEM Busness Coputng nd Opertons Reserch 774

16 he RP s specfc Flow Proble { } ( RP) Obvously, the proble cn be odeled s Mx-Flow Proble. For ths purpose, we defne the followng network: V s,v,...,v,w,...,w,t { } { } (, ) {( w ) },t n c s,v, n E s,v v,w n IJ {,..., } (, ), (, ) (, ), {,..., } c w t b n c v w IJ Busness Coputng nd Opertons Reserch 775

17 Illustrton of the network v w v IJ b w b s v w b t... Edges re out of set IJ nd hve unlted cpcty... b n v w n Busness Coputng nd Opertons Reserch 776

18 Resung wth our exple In the exple ntroduced bove, we generted the followng ntl soluton α β hus, we cn derve β Wth α we obtn the reduced trx: 4 IJ (, ),(,4 ),(, ),(, ),(,4) { } Busness Coputng nd Opertons Reserch 777

19 We obtn the followng network v w w 5 s v 6 t 6 w v w 4 Busness Coputng nd Opertons Reserch 778

20 Augentng the flow At frst, we fnd the flow s-v -w -t It cn be ugented up to herefore, we updte the network Busness Coputng nd Opertons Reserch 779

21 We obtn the odfed network v w w 5 s v t 6 w v w 4 Busness Coputng nd Opertons Reserch 78

22 Augentng the flow Now, we fnd s-v -w -t It cn be ugented up to herefore, we updte the network Busness Coputng nd Opertons Reserch 78

23 Illustrton v w s v w t 6 w v w 4 Busness Coputng nd Opertons Reserch 78

24 Augentng the flow Now, we fnd s-v -w -t It cn be ugented up to herefore, we updte the network Busness Coputng nd Opertons Reserch 78

25 Modfyng our network gn v w 5 s v w t 6 w v w 4 Busness Coputng nd Opertons Reserch 784

26 Augentng gn the flow Now, we fnd s-v -w 4 -t It cn be ugented up to herefore, we updte the network Busness Coputng nd Opertons Reserch 785

27 And the network s dusted to v w 5 s v w w t v w 4 Busness Coputng nd Opertons Reserch 786

28 Soluton to the reduced prl proble hus, x Owng to the vectors b vrbles we obtn : ( 6 ) x., Obvously ( 5 6) ( ) x s we need the vector of not fesble for slckness ( P) Busness Coputng nd Opertons Reserch 787

29 Updtng the dul soluton Obvously, we cn optlly solve (RP) by kng use of n effcent Mx-Flow Algorth Unfortuntely, ths does not provde echns for updtng the dul soluton yet In order to do so, we hve to nlyze the dul of the reduced prl (DRP) Busness Coputng nd Opertons Reserch 788

30 Modfed Reduced Prl (RP ) Mnze x, s.t., n+ x,, x,,..., n +,,, IJ +,, IJ { } x + x,,..., { } x + x b,,..., n { } Busness Coputng nd Opertons Reserch 789

31 Modfed Reduced Prl (RP ) Snce t holds n n x + x, b x + + x,, IJ, IJ n n x x, x + x,, IJ, IJ + + n n + n x x, x + x, x x + x x, IJ, IJ + hus, we obtn the equvlent proble: { } { } Mnze x, s.t., x,, x,,..., n +,, +,, IJ, IJ { } x + x,,..., x + x b,,..., n Busness Coputng nd Opertons Reserch 79

32 nd ts dul counterprt (DRP ) Mxze s.t., α + β,, IJ α,,..., β,,..., n α + b β n { } { } Busness Coputng nd Opertons Reserch 79

33 8. Solvng the DRP 8.. heore ( RP) Assung ws optlly solved by n pproprte ( c W,W ) Mx-Flow Algorth. Furtherore, s the resultng s - t - cut ccordng to the current x wth { s rechble fro n the fnl network of } W v V v s RP hen, αɶ f v f W w W βɶ c c f v f w W W ( DRP ) ( ˆ ˆ ) deternes n optl soluton for. Addtonlly, α, β, wth ˆ α α + λ ɶ α. ( D) ˆ β β + λ ɶ β re proved solutons of f n+ x > λ > Busness Coputng nd Opertons Reserch 79

34 Proof of the heore Bsc cogntons As prelnry step, we generte soe bsc ttrbutes. If v W, we know tht: f ddtonlly, IJ w W hs results fro the followng observton: If v W, IJ, then we know tht there s n edge wth unlted cpcty connectng v nd w. Hence, t holds c > f nd therefore,, w s rechble fro s s well. Busness Coputng nd Opertons Reserch 79

35 Proof of the heore Bsc cogntons. Corollry: c v W w W, IJ,. If w W, IJ x > v W, hs results fro the followng observton: Snce x >, forer step hs estblshed connecton between v nd. hus we hve bckwrd lnk fro w to v wth cpcty x >., w Busness Coputng nd Opertons Reserch 794

36 Proof of the heore Bsc cogntons (, ) 4. Corollry: v W w W, IJ x c, In wht follows, r denotes the renng cpcty on the lnk,,wth { } { } { } { } { }, v,w, IJ s,v,..., w,t,...,n. 5. v W r x x c s, v,, E 6. w W r x b x + w, t,, E Busness Coputng nd Opertons Reserch 795

37 Proof of heore 8.. Fesblty We re now redy to coence the proof. At frst, we show the ( DRP) fesblty of the generted soluton to. Obvously, t holds: { } ɶ { }. α ɶ,,..., β,,...,n Addtonlly, we hve to show. α + βɶ,, IJ. ɶ. v W w W αɶ βɶ αɶ + βɶ c. v W αɶ αɶ + βɶ ( αɶ ) βɶ ( DRP) hus,, s fesble soluton to. Busness Coputng nd Opertons Reserch 796

38 Proof of heore 8.. Optlty We know tht the optl soluton to the reduced prl proble s generted by the Mx-Flow procedure nd s therefore defned by the followng vrbles, { } x,, IJ x,,..., n + Consequently, ts obectve functon vlue s deterned by We clculte: αɶ + b βɶ b v W w W x + x x + x,, + v W, IJ v W w W, IJ w W n x Busness Coputng nd Opertons Reserch 797

39 RP nd DRP hve dentcl obectve vlues And thus, t holds: n α + b β x + x x x,, + v W, IJ v W w W, IJ w W x x + x x x x,, + +, IJ, IJ v W w W v W w W x v W x Owng to ttrbute 6, ths s equl to ( α, ɶ βɶ ) ( DRP ) hus, s n optl soluton to Busness Coputng nd Opertons Reserch 798

40 Fesblty of the updted dul soluton We clculte ( ɶ ) ( α,β ˆ ˆ ) ( α,β) + λ α,β ɶ ɶ It hs to be gurnteed α + λ αɶ + β + λ βɶ c α + β + λ αɶ + λ βɶ c,, λ α + βɶ c α β λ, f v W w W c c f v W w W α + βɶ ɶ f v W w W c f v W w W c α β αɶ + βɶ, c f v W w W c f v W w W otherwse c Busness Coputng nd Opertons Reserch 799

41 Proof of heore 8.. Defnng λ c f v W w W c, α β c λ, wth αɶ + βɶ f v W w W αɶ + βɶ otherwse If c, IJ, we hve to consder the cse v W w W { c }, λ n α + β c, IJ v W w W If, IJ, we hve to consder the cse v W w W,, { } λ n c α β, IJ > hus, we defne { c} λ n c α β, IJ v W w W > c Busness Coputng nd Opertons Reserch 8

42 Busness Coputng nd Opertons Reserch 8 Qulty of the new dul soluton { } > + > > n x n n n n n β b α x λ β b α β b α λ β b α β b λ β b α λ α β λ β b α λ α IJ, β α c λ If, ~ ~ ~ ~ ~ ~ we clculte, n Wth

43 And wht follows? Gret! ht s ll we need to optlly solve the proble However, we y splfy the forul consderbly... Busness Coputng nd Opertons Reserch 8

44 Iportnt observton Prt We consder the resultng constellton fter pplyng the Mx-Flow procedure. Addonlly, we nlyze the generted flow. Frst of ll, we consder rcs tht vnsh n the next terton. hs y hppen only f, IJ n the current terton, but n the next one t holds, IJ. hs cse s chrcterzed tht orgnlly α + β c pples, but subsequently αˆ + βˆ < c holds. Note,, tht ths s only possble f αɶ + βɶ < αɶ + βɶ. hs s the constellton x, c v W w W. It s llustrted on the next slde. Here, we drectly conclude tht the rc, IJ ws not used by the generted flow t ll. Hence, we obtn x., Busness Coputng nd Opertons Reserch 8

45 Illustrton of ths constellton v w b v w b s v W c w b t sturted rc... unused rc W... b n v w n Busness Coputng nd Opertons Reserch 84

46 Consequence If we erse the edge (,) n the subsequent terton,.e., the solvng of the odfed (RP), ths hs no pct on the current flow x, Note tht the current flow does not ke use of ths rc Consequently, ths rc s dspensble Busness Coputng nd Opertons Reserch 85

47 Observtons II, (, ) IJ x, Now we consder rcs wth >. We know tht t holds αˆ + βˆ c αɶ + βɶ. herefore, the flow, x > cn be kept on these rcs. Anyhow, the resultng flow x cn be kept for the next ( RP) terton of solvng tht rses fter updtng α nd, β. Note tht ths updte y cuse ddtonl rcs between the v nd w nodes. Busness Coputng nd Opertons Reserch 86

48 , Clcultng λ { c} λ n c α β, IJ v W w W hus, we cn lbel ll rows n the reduced trx c α β wth v W, coluns wth w W. c. Addtonlly, we lbel ll hen λ s deterned by the nu unlbeled vlue. ( c ˆ ˆ ), α β We updte by pplyng the followng rules: Busness Coputng nd Opertons Reserch 87

49 Updtng rules We dstngush:. If, s unlbeled v W w W We subtrct λ fro c α β c. If, s lbeled twce v W w W α + β. We dd λ to c α β,,. If, s lbeled only by the th row or the th colun ( c c ) v W w W v W w W α+ β c α β, s kept unchnged c Busness Coputng nd Opertons Reserch 88

50 Contnuton of the exple Now, we resue our exple whch ws ntroduced bove hus, frst of ll, we hve to updte the dul soluton Wth β ( ) Reduced trx s therefore: IJ α {(, ), (,4 ), (, ), (,),(,4) } 4 Busness Coputng nd Opertons Reserch 89

51 Illustrton of the clculton W v w 5 s v v w w t W c w 4 Busness Coputng nd Opertons Reserch 8

52 Updtng the dul soluton W s,v,w W c { } 4 v, v, w, w, w, t { } Wth α β ( c ), α β 4 4 Busness Coputng nd Opertons Reserch 8

53 λ Updtng the dul soluton + n{,, 4} α ( ) β ( ) ( ) βɶ ( ) ˆ β (, ),(, ),(, ),(, ),(, ),(,4) α αˆ ɶ hus, we get two new rcs, nd, nd lose one,4. IJ { } Busness Coputng nd Opertons Reserch 8

54 Illustrton v w w 5 s v w t v w 4 Busness Coputng nd Opertons Reserch 8

55 Applyng Mx-Flow v w w 5 s v w t v w 4 Busness Coputng nd Opertons Reserch 84

56 Results Unfortuntely, we re not ble to ugent the flow hus, x s kept s xu flow However, we hve chnged the sets W nd W c hs s consdered n the followng Busness Coputng nd Opertons Reserch 85

57 Applyng Mx-Flow W v W c w w 5 s v w t v w 4 Busness Coputng nd Opertons Reserch 86

58 Updtng the dul soluton W W c s,v, v, w, w, w { } 4 v, w, t { } Wth α β ( c ), α β Busness Coputng nd Opertons Reserch 87

59 λ α Updtng the dul soluton n{,} β αɶ βɶ αˆ 4 βˆ hus, we get new rcs,. IJ {(, ),(, ),(, ),(,)(, ),(, ),(,4) } Busness Coputng nd Opertons Reserch 88

60 Modfed network v w w 5 s v w t v w 4 Busness Coputng nd Opertons Reserch 89

61 We obtn the ugented flow v w w 5 s v w t v w 4 Busness Coputng nd Opertons Reserch 8

62 Illustrton v w w 5 s v t 6 6 w v w 4 Busness Coputng nd Opertons Reserch 8

63 he new decoposton W v w W c w 5 s v t 6 6 w v w 4 Busness Coputng nd Opertons Reserch 8

64 Busness Coputng nd Opertons Reserch 8 he odfed prl soluton { } { } Is fesble for 4 Wth,,,,,,, 4 c b x β α t w w w w v v v W s, W

65 Proof of optlty W s, W v, v, v, w, w, w, w, t { } c { } { } 4 x,,..., + n nd t holds: c x α + b β x nd α, β re optl solutons! Busness Coputng nd Opertons Reserch 84

66 Alph-Bet-Algorth. Construct fesble dul soluton to the PP n c,..., α β Set β { } nd { } n c,..., n Clculte the trx wth the reduced costs c c α β. Prepre the network for the Mx-Flow-Clculton Nodes: Arcs: s, v,..., v, w,..., wn, t s v s v wth cpcty,, w t wn t b,, bn (, ),...,(, ) (, ),...,(, ). Furtherore: If nd only f c, the rc ( v, w ) exsts wth nfnte cpcty 4. Clculte the Mxu s-t-flow n the network. Let W be the set of nodes rechble fro node s n the correspondng s-t-cut { } 5. Whle W s, conduct the followng steps (see next slde): Busness Coputng nd Opertons Reserch 85

67 Alph-Bet-Algorth (Dul Soluton Updte) c If v W ɶ α ; v W, lbel the -th row n the reduced cost trx. ɶ If w W β, lbel the -th colun n the reduced cost trx. All other vrbles of the DRP-soluton ɶ α, ɶ β re set to. λ Set to the nu vlue of the unlbeled entres n the reduced cost trx. λ Subtrct fro every unlbeled entry nd dd t to every entry lbeled twce n the reduced cost trx. Set β β + λ ɶ β α α + λαɶ Updte the network s ndcted by the new reduced cost trx. ry to ugent the current flow nd updte the set W. Busness Coputng nd Opertons Reserch 86