D. L. Brcker, 2002 Dept of Mechacal & Idustral Egeerg The Uversty of Iowa CPL/XD 2/0/2003 page
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
D. L. Brcker Dept of Mechacal & Idustral Egeerg The Uversty of Iowa CPL/XD Exaple 05/29/02 page of 7
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 3 49 7 4 9 43 8 7 46 82 3 0 87 26 3 88 63 8 2 44 94 5 5 46 89 7 8 87 5 8 5 3 4 66 25 2 3 24 39 2 6 47 4 9 9 35 37 2 2 64 42 8 Total dead: 70 CPL/XD Exaple 05/29/02 page 2 of 7
Pots 2 3 4 5 6 7 are potetal plat stes, wth capactes & fxed costs K F 37 405 2 63 344 3 25 330 4 59 6 5 82 292 6 48 498 7 95 28 ( = plat ste #, K[] = capacty, F[] = fxed cost) K = capacty, F = fxed cost Costs, Supples, & Deads: /j 2 3 4 5 6 7 8 9 0 2 3 4 K F 0 6 23 8 59 45 54 97 34 87 46 6 86 67 37 405 2 6 0 59 62 5 53 2 02 58 80 93 56 54 72 63 344 3 23 59 0 6 55 23 48 74 64 35 40 68 44 25 330 4 8 62 6 0 59 38 54 89 27 80 39 55 8 60 59 6 5 59 5 55 59 0 48 7 97 53 75 89 50 49 67 82 292 6 45 53 23 38 48 0 4 57 3 43 46 7 47 25 48 498 7 54 2 48 54 7 4 0 9 46 69 82 44 46 60 95 28 Dead: 7 5 2 8 7 9 3 5 2 3 8 8 2 409 0 CPL/XD Exaple 05/29/02 page 3 of 7
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
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
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 < 2 3 5 > Estated V(X): 775 Iterato #4 Tral Y for pral subprobles: ope # 2 3 5 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
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 < 2 5 6 7 > Estated V(X): 544 Iterato #6 Tral Y for pral subprobles: ope # 2 5 6 7 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
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 <2 3 4 6 > Estated V(X): 740 Iterato # Tral Y for pral subprobles: ope #2 3 4 6 Pral subproble results: Trasport costs 44 Fxed costs 288 Total costs 2432 Soluto of Master Proble Y: ope <3 4 5 6 7 > Estated V(X): 765 Iterato #2 Tral Y for pral subprobles: ope #3 4 5 6 7 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
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 <2 3 4 7 > Estated V(X): 979 Iterato #4 Tral Y for pral subprobles: ope #2 3 4 7 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 <2 4 6 7 > Estated V(X): 204 Iterato #6 Tral Y for pral subprobles: ope #2 4 6 7 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
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 4 6 59 39 5 292 82 67 6 498 48 3 Total fxed costs= 906 = 43.70% of total cost CPL/XD Exaple 05/29/02 page 0 of 7
Optal Shpets 2 3 4 5 6 7 8 9 0 2 3 4 5 4 7 0 2 8 0 0 0 0 0 0 3 0 0 0 39 5 0 5 0 0 7 0 3 0 0 0 0 0 0 0 67 6 0 0 0 0 0 9 0 5 2 0 8 8 2 3 (Dead pt #5 s duy dead for excess capacty.) Supply costrats U U 8 4 6 2 5 6 3 0 7 9 Dual Soluto of Trasportato Proble Dead costrats j Vj j Vj j Vj j Vj j Vj j Vj j Vj 8 3 0 5 6 7 9 9 3 23 3 3 2 4 6 6 6 8 4 0 27 2 4 9 CPL/XD Exaple 05/29/02 page of 7
Reduced costs: COST - U.+V I \ J= 2 3 4 5 6 7 8 9 0 2 3 4 0 64 5 6 67 53 55 48 29 52 5 52 47 50 2 58 0 48 67 0 58 0 50 50 42 59 44 2 52 3 3 70 0 32 7 39 57 33 4 37 2 39 37 35 4 0 57 0 0 59 38 47 32 4 37 0 38 34 35 5 5 0 39 59 0 48 0 40 40 32 50 33 2 42 6 37 48 7 38 48 0 34 0 0 0 7 0 0 0 7 53 4 39 6 4 48 0 4 40 33 50 34 6 42 CPL/XD Exaple 05/29/02 page 2 of 7
Lower boud s ootocally creasg! CPL/XD Exaple 05/29/02 page 3 of 7
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
Iterato # Dual subproble results: Usg ultplers: 2 3 4 5 6 7 Mu[] 2.62.69 2.96.508.82.87 0.957 Subproble X: Optal cost= 009, X= 2 3 4 5 6 7 8 9 0 2 3 4 su 7 0 0 0 0 0 0 0 0 0 0 0 0 0 7 2 0 5 0 0 0 0 0 0 0 0 0 0 0 0 5 3 0 0 2 0 0 0 0 0 2 0 3 0 0 0 7 4 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 5 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 6 0 0 0 0 0 9 0 5 0 0 8 0 2 25 7 0 0 0 0 0 0 3 0 0 0 0 0 8 0 Subproble Y: Objectve coeffcets: 2 3 4 5 6 7 cost 308 242 256 27 95 44 90 Optal cost= 697, by opeg plats 3 6 Total cost (Lower boud): 706 ---------------------------------------- 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: 30 0 53 37 5 53 2 *** ew cubet! *** Iterato #2 Dual subproble results: Usg ultplers: 2 3 4 5 6 7 Mu[] 30 0 53 37 5 53 2 Subproble X: Optal cost= 2222, X= 2 3 4 5 6 7 8 9 0 2 3 4 su 7 0 2 0 0 0 0 0 0 0 3 0 0 0 2 2 0 5 0 0 7 0 0 5 0 0 0 8 0 26 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 5 0 0 0 0 0 0 0 0 0 0 0 8 0 0 8 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 9 3 0 2 0 0 0 0 2 6 Subproble Y: Objectve coeffcets: 2 3 4 5 6 7 cost 705 344 995 2067 8 2046 859 Optal cost= 6790, by opeg plats 3 4 5 6 7 Total cost (Lower boud): 4568 ---------------------------------------- Pral Subproble Tral Y for pral subproble s: ope plats # 3 4 5 6 7 wth fxed costs 922 Pral subproble soluto: Trasportato cost = 034 Total cost = 2956 Dual varables: 5 0 5 5 5 5 5 CPL/XD Exaple 05/29/02 page 5 of 7
Iterato #3 Dual subproble results: Usg ultplers: 2 3 4 5 6 7 Mu[] 5 0 5 5 5 5 5 Subproble X: Optal cost= 044, X= 2 3 4 5 6 7 8 9 0 2 3 4 su 7 0 0 0 0 0 0 0 0 0 0 0 0 0 7 2 0 5 0 0 7 0 0 0 0 0 0 0 0 0 2 3 0 0 2 0 0 0 0 0 2 0 3 0 0 0 7 4 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 9 0 5 0 0 8 0 2 25 7 0 0 0 0 0 0 3 0 0 0 0 0 8 0 Subproble Y: Objectve coeffcets: 2 3 4 5 6 7 cost 220 344 205 79 8 258 94 Optal cost= 49, by opeg plats 4 5 7 Total cost (Lower boud): 553 ---------------------------------------- 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: 30 33 22 38 38 0 38 *** ew cubet! *** Iterato #4 Dual subproble results: Usg ultplers: 2 3 4 5 6 7 Mu[] 30 33 22 38 38 0 38 Subproble X: Optal cost= 53, X= 2 3 4 5 6 7 8 9 0 2 3 4 su 7 0 0 8 0 0 0 0 0 0 0 0 0 0 5 2 0 5 0 0 7 0 0 0 0 0 0 0 0 0 2 3 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 9 0 5 2 3 8 8 2 38 7 0 0 0 0 0 0 3 0 0 0 0 0 0 0 3 Subproble Y: Objectve coeffcets: 2 3 4 5 6 7 cost 705 735 220 226 2824 498 3329 Optal cost= 0939, by opeg plats 2 3 4 5 7 Total cost (Lower boud): 9786 ---------------------------------------- Pral Subproble Tral Y for pral subproble s: ope plats # 2 3 4 5 7 wth fxed costs 768 Pral subproble soluto: Trasportato cost = 572 Total cost = 3340 Dual varables: 27 27 23 27 27 0 27 CPL/XD Exaple 05/29/02 page 6 of 7
Covergece occurs terato #29: CPL/XD Exaple 05/29/02 page 7 of 7