D. L. Bricker, 2002 Dept of Mechanical & Industrial Engineering The University of Iowa. CPL/XD 12/10/2003 page 1

Transcription

1 D. L. Brcker, 2002 Dept of Mechacal & Idustral Egeerg The Uversty of Iowa CPL/XD 2/0/2003 page

2 Capactated Plat Locato Proble: Mze FY + C X subject to = = j= where Y = j= X D, j =, j X SY, =,... X 0, =, ; j =, { } Y 0,, =, f a plat s bult at ste = 0 otherwse X = quatty suppled by plat at ste to custoer j CPL/XD 2/0/2003 page 2

3 The followg addtoal costrats are redudat but are potetally useful, depedg upo how we choose the Lagraga relaxato: X D Y, =, ; j =, j These costrats, together wth X S, =,..., j= could be used stead of the costrats j= X SY, =,... order to force shpets fro a plat to be zero f that plat has ot bee opeed! CPL/XD 2/0/2003 page 3

4 The costrat SY j = j= D s redudat, but s useful order to geerate tral solutos (Y) whch are guarateed to gve feasble subprobles! CPL/XD 2/0/2003 page 4

5 A alterate forulato of the CPL: Mze FY + C X subject to = = j= = j= X D, j =, j X S, =, SY j = j= D X D Y, =, ; j =, (lkg costrats) j X 0,, ; j, = =, { } Y 0,, =, CPL/XD 2/0/2003 page 5

6 Our goal s to separate the proble by relaxg the costrats lkg the X ad Y decsos. A Lagraga Relaxato of the CPL: ( ) D µ = Mu FY + C X + µ X D Y subject to j = = j= = j= = j= X D, j =, j X S, =, SY j = j= D X 0, =, ; j =, { } Y 0,, =, CPL/XD 2/0/2003 page 6

7 Rearragg ters the objectve fucto: SY j = j= { } D ( µ ) = ( µ ) + ( + µ ) D Mu F D Y C X subject to Y 0,, =, j = = j= = j= X D, j =, j X S, =, X 0, =, ; j =, CPL/XD 2/0/2003 page 7

8 Ths separates to two subprobles: D( µ ) = D ( µ ) + D ( µ ).e., the trasportato proble ( µ ) = ( + µ ) D Mu C X X = j= subject to = X D, j =, j X Y j= X S, =, ad the proble: X 0, =, ; j =, ( µ ) = ( µ ) D Mu F D Y Y j = j = j= { } subject to S Y D, Y 0, CPL/XD 2/0/2003 page 8

9 By usg the copleets of the varables,.e., Y Y, the subproble D Y ( µ ) ca be expressed as a 0- kapsack proble: ( µ ) = ( µ ) Maxu ( µ ) D F D F D Y Y j j = = j = = j= { } subject to SY S D, Y 0, CPL/XD 2/0/2003 page 9

10 Cosder stll aother forulato of CPL: j= Mze FY + C X subject to = j= = = j= X D, j =, j X S, =, SY j = j= X SY, =, (lkg costrats) D X 0, =, ; j =, { } Y 0,, =, CPL/XD 2/0/2003 page 0

11 A Lagraga Relaxato of the CPL, where µ 0: D( µ ) = Mu FY + CX + µ X SY = = j= = j= or ( µ ) = ( µ ) + ( + µ ) D Mu F S Y C X subject to = j= = = j= X D, j =, j X S, =, SY j = j= D X 0, =, ; j =, { } Y 0,, =, CPL/XD 2/0/2003 page

12 Ths separates to two subprobles: D( µ ) = D ( µ ) + D ( µ ).e., the trasportato proble ( µ ) = ( + µ ) D Mu C X X = j= subject to = X D, j =, j X Y j= X S, =, ad the proble: X 0, =, ; j =, ( µ ) = ( µ ) D Mu F S Y Y = j = j= { } subject to S Y D, Y 0, CPL/XD 2/0/2003 page 2

13 As before, by usg the copleets of the varables,.e., Y ( ) Y Y, D µ ca be expressed as a 0- kapsack proble: ( µ ) = ( µ ) Maxu ( µ ) D F S S F Y Y = = j = = j= { } subject to SY S D, Y 0, CPL/XD 2/0/2003 page 3

14 Cross-Decoposto Ether Lagraga subproble ca be used to geerate tral solutos (Y) for the Beders' subprobles (whch tur geerates Lagraga ultplers for the Lagraga subprobles) a Cross-Decoposto schee! Note that f lower bouds are ot requred, oly the Lagraga subproble the Y varables (.e., the kapsack proble) eeds to be solved at each terato to provde the tral soluto for Beders' subproble CPL/XD 2/0/2003 page 4

15 STALLING "Stallg" ca occur,.e., the sae tral Y varables or the sae dual varables ay be geerated ultple tes, whch case t s ecessary to resort to the Beders' (or Lagraga) Master Proble. I order to avod ths, ea values ay be passed fro the pral subproble to dual subproble, ad/or vce versa. CPL/XD 2/0/2003 page 5

16 Note: Cuts or costrats for Beders Master Proble ay be derved as follows. Assue that the subproble s feasble,.e., SY j = j= D Trasfor the supply & dead costrats to equatos by defg a duy dead pot (#+) : + FY + Mze C X X 0 = = j= subject to = X = D, j =, j X = SY, =,... j= X = SY D, + j = = j= CPL/XD 2/0/2003 page 6

17 Let U & V be the optal dual varables. The the optal value v(y) s FY + U SY + V D + V SY D j j + j = = j= = j= ( + ) ( + ) = U + V SY + V V D j j = j= Thus, the lear (uder-)estate of the optal value fucto v(y) s α Y +β, where ( ) α = U + V S, =, + ( + ) β= V V D j= j j CPL/XD 2/0/2003 page 7

18 D. L. Brcker Dept of Mechacal & Idustral Egeerg The Uversty of Iowa CPL/XD Exaple 05/29/02 page of 7

19 Radoly-geerated proble wth 7 potetal plat stes ad 4 dead pots Rado uber seed = 3432 X Y D X Y D X Y D X Y D X Y D Total dead: 70 CPL/XD Exaple 05/29/02 page 2 of 7

20 Pots are potetal plat stes, wth capactes & fxed costs K F ( = plat ste #, K[] = capacty, F[] = fxed cost) K = capacty, F = fxed cost Costs, Supples, & Deads: /j K F Dead: CPL/XD Exaple 05/29/02 page 3 of 7

21 We solve ths proble by Stadard Beders decoposto (optzg aster proble at each terato) Stadard Cross-Decoposto CPL/XD Exaple 05/29/02 page 4 of 7

22 Frst the proble s solved by Beders decoposto algorth: Beders Decoposto Algorth Master proble wll be optzed at each terato, provdg the Y zg curret approxato v(y) ad a lower boud CPL/XD Exaple 05/29/02 page 5 of 7

23 Iterato # Tral Y for pral subprobles: ope #3 6 (tal guess ) Pral subproble results: Trasport costs 2030 Fxed costs 828 Total costs 2858 **** New cubet! **** Soluto of Master Proble Y: ope < 2 7 > Estated V(X): 4579 Iterato #2 Tral Y for pral subprobles: ope #2 7 Pral subproble results: Trasport costs 302 Fxed costs 625 Total costs 3637 Soluto of Master Proble Y: ope <4 6 7 > Estated V(X): 247 Iterato #3 Tral Y for pral subprobles: ope #4 6 7 Pral subproble results: Trasport costs 222 Fxed costs 895 Total costs 27 **** New cubet! **** Soluto of Master Proble Y: ope < > Estated V(X): 775 Iterato #4 Tral Y for pral subprobles: ope # Pral subproble results: Trasport costs 723 Fxed costs 37 Total costs 3094 Soluto of Master Proble Y: ope <2 3 6 > Estated V(X): 3 CPL/XD Exaple 05/29/02 page 6 of 7

24 Iterato #5 Tral Y for pral subprobles: ope #2 3 6 Pral subproble results: Trasport costs 377 Fxed costs 72 Total costs 2549 Soluto of Master Proble Y: ope < > Estated V(X): 544 Iterato #6 Tral Y for pral subprobles: ope # Pral subproble results: Trasport costs 56 Fxed costs 820 Total costs 2976 Soluto of Master Proble Y: ope <4 5 6 > Estated V(X): 590 Iterato #7 Tral Y for pral subprobles: ope #4 5 6 Pral subproble results: Trasport costs 67 Fxed costs 906 Total costs 2073 **** New cubet! **** Soluto of Master Proble Y: ope <4 7 > Estated V(X): 69 Iterato #8 Tral Y for pral subprobles: ope #4 7 Pral subproble results: Trasport costs 2064 Fxed costs 397 Total costs 246 Soluto of Master Proble Y: ope <4 6 > Estated V(X): 836 CPL/XD Exaple 05/29/02 page 7 of 7

25 Iterato #9 Tral Y for pral subprobles: ope #4 6 Pral subproble results: Trasport costs 863 Fxed costs 64 Total costs 2477 Soluto of Master Proble Y: ope <4 5 > Estated V(X): 909 Iterato #0 Tral Y for pral subprobles: ope #4 5 Pral subproble results: Trasport costs 2079 Fxed costs 408 Total costs 2487 Soluto of Master Proble Y: ope < > Estated V(X): 740 Iterato # Tral Y for pral subprobles: ope # Pral subproble results: Trasport costs 44 Fxed costs 288 Total costs 2432 Soluto of Master Proble Y: ope < > Estated V(X): 765 Iterato #2 Tral Y for pral subprobles: ope # Pral subproble results: Trasport costs 090 Fxed costs 57 Total costs 2607 Soluto of Master Proble Y: ope <5 6 > Estated V(X): 957 CPL/XD Exaple 05/29/02 page 8 of 7

26 Iterato #3 Tral Y for pral subprobles: ope #5 6 Pral subproble results: Trasport costs 779 Fxed costs 790 Total costs 2569 Soluto of Master Proble Y: ope < > Estated V(X): 979 Iterato #4 Tral Y for pral subprobles: ope # Pral subproble results: Trasport costs 663 Fxed costs 07 Total costs 2734 Soluto of Master Proble Y: ope <6 7 > Estated V(X): 200 Iterato #5 Tral Y for pral subprobles: ope #6 7 Pral subproble results: Trasport costs 820 Fxed costs 779 Total costs 2599 Soluto of Master Proble Y: ope < > Estated V(X): 204 Iterato #6 Tral Y for pral subprobles: ope # Pral subproble results: Trasport costs 48 Fxed costs 239 Total costs 2387 Soluto of Master Proble Coverged at terato #6! o tral soluto CPL/XD Exaple 05/29/02 page 9 of 7

27 Rado Proble (Seed = 3432) (Foud at terato #7!) Icubet Soluto Suary Trasport cost= 67 Fxed costs= 906 Total costs= 2073 Low boud 2073 Gap (%) 0 Plat Fxed Cost Supply Surplus Total fxed costs= 906 = 43.70% of total cost CPL/XD Exaple 05/29/02 page 0 of 7

28 Optal Shpets (Dead pt #5 s duy dead for excess capacty.) Supply costrats U U Dual Soluto of Trasportato Proble Dead costrats j Vj j Vj j Vj j Vj j Vj j Vj j Vj CPL/XD Exaple 05/29/02 page of 7

29 Reduced costs: COST - U.+V I \ J= CPL/XD Exaple 05/29/02 page 2 of 7

30 Lower boud s ootocally creasg! CPL/XD Exaple 05/29/02 page 3 of 7

31 Cross-Decoposto Algorth Curret paraeters for cross-decoposto Method for geeratg Y for pral subprobles Most recet Method for updatg Lagraga ultplers for use dual subprobles Most recet CPL/XD Exaple 05/29/02 page 4 of 7

32 Iterato # Dual subproble results: Usg ultplers: Mu[] Subproble X: Optal cost= 009, X= su Subproble Y: Objectve coeffcets: cost Optal cost= 697, by opeg plats 3 6 Total cost (Lower boud): Pral Subproble Tral Y for pral subproble s: ope plats #3 6 wth fxed costs 828 Pral subproble soluto: Trasportato cost = 2030 Total cost = 2858 Dual varables: *** ew cubet! *** Iterato #2 Dual subproble results: Usg ultplers: Mu[] Subproble X: Optal cost= 2222, X= su Subproble Y: Objectve coeffcets: cost Optal cost= 6790, by opeg plats Total cost (Lower boud): Pral Subproble Tral Y for pral subproble s: ope plats # wth fxed costs 922 Pral subproble soluto: Trasportato cost = 034 Total cost = 2956 Dual varables: CPL/XD Exaple 05/29/02 page 5 of 7

33 Iterato #3 Dual subproble results: Usg ultplers: Mu[] Subproble X: Optal cost= 044, X= su Subproble Y: Objectve coeffcets: cost Optal cost= 49, by opeg plats Total cost (Lower boud): Pral Subproble Tral Y for pral subproble s: ope plats #4 5 7 wth fxed costs 689 Pral subproble soluto: Trasportato cost = 980 Total cost = 2669 Dual varables: *** ew cubet! *** Iterato #4 Dual subproble results: Usg ultplers: Mu[] Subproble X: Optal cost= 53, X= su Subproble Y: Objectve coeffcets: cost Optal cost= 0939, by opeg plats Total cost (Lower boud): Pral Subproble Tral Y for pral subproble s: ope plats # wth fxed costs 768 Pral subproble soluto: Trasportato cost = 572 Total cost = 3340 Dual varables: CPL/XD Exaple 05/29/02 page 6 of 7

34 Covergece occurs terato #29: CPL/XD Exaple 05/29/02 page 7 of 7