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1 . L. Brcker, 2002 ept of Mechacal & Idustral Egeerg The Uversty of Iowa CPL/ 5/29/2002 page CPL/ 5/29/2002 page 2 Capactated Plat Locato Proble: where Mze F + C subect to = = =, =, S, =,... 0, =, ; =, 0,, =, f a plat s bult at ste = 0 otherwse = quatty suppled by plat at ste to custoer The followg addtoal costrats are redudat but are potetally useful, depedg upo how we choose the Lagraga relaxato:, =, ;, These costrats, together wth S, =,..., could be used stead of the costrats S, =,... order to force shpets fro a plat to be zero f that plat has ot bee opeed! CPL/ 5/29/2002 page 3 CPL/ 5/29/2002 page 4
2 The costrat = s redudat, but s useful order to geerate tral solutos ( whch are guarateed to gve feasble subprobles! A alterate forulato of the CPL: Mze F + C subect to = = =, =, S, =, =, =, ;, (lkg costrats 0,, ;, = =, 0,, =, CPL/ 5/29/2002 page 5 CPL/ 5/29/2002 page 6 Our goal s to separate the proble by relaxg the costrats lkg the ad decsos. A Lagraga Relaxato of the CPL: ( µ = Mu F + C + µ subect to = = = Rearragg ters the obectve fucto: = ( µ = ( µ + ( + µ Mu F C subect to 0,, =, = = =, =, S, =, 0, =, ; =, =, =, S, =, = 0, =, ; =, 0,, =, CPL/ 5/29/2002 page 7 CPL/ 5/29/2002 page 8
3 Ths separates to two subprobles: ( µ = ( µ + ( µ.e., the trasportato proble ( µ = ( + µ Mu C = subect to =, =, S, =, 0, =, ; =, By usg the copleets of the varables,.e.,, the subproble ( µ ca be expressed as a 0- kapsack proble: ( µ = ( µ Maxu ( µ F F = = S S = = subect to, 0, ad the proble: ( µ = ( µ Mu F = = subect to S, 0, CPL/ 5/29/2002 page 9 CPL/ 5/29/2002 page 0 Cosder stll aother forulato of CPL: Mze F + C subect to = = =, =, S, =, = S, =, (lkg costrats 0, =, ; =, 0,, =, A Lagraga Relaxato of the CPL, where µ 0: ( µ = Mu F + C + µ S = = = or ( µ = ( µ + ( + µ Mu F S C subect to = = =, =, S, =, = 0, =, ; =, 0,, =, CPL/ 5/29/2002 page CPL/ 5/29/2002 page 2
4 Ths separates to two subprobles: ( µ = ( µ + ( µ.e., the trasportato proble ( µ = ( + µ Mu C = subect to =, =, S, =, 0, =, ; =, As before, by usg the copleets of the varables,.e., (, µ ca be expressed as a 0- kapsack proble: ( µ = ( µ Maxu ( µ F S S F = = = = subect to S S, 0, ad the proble: ( µ = ( µ Mu F S = = subect to S, 0, CPL/ 5/29/2002 page 3 CPL/ 5/29/2002 page 4 Cross-ecoposto Ether Lagraga subproble ca be used to geerate tral solutos ( for the Beders' subprobles (whch tur geerates Lagraga ultplers for the Lagraga subprobles a Cross-ecoposto schee! Note that f lower bouds are ot requred, oly the Lagraga subproble the varables (.e., the kapsack proble eeds to be solved at each terato to provde the tral soluto for Beders' subproble STALLING "Stallg" ca occur,.e., the sae tral 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/ 5/29/2002 page 5 CPL/ 5/29/2002 page 6
5 Note: Cuts or costrats for Beders Master Proble ay be derved as follows. Assue that the subproble s feasble,.e., = Trasfor the supply & dead costrats to equatos by defg a duy dead pot (#+ : Let U & V be the optal dual varables. The the optal value v( s F + U S + V + V S + = = = ( + ( + = U + V S + V V = + F + Mze C 0 = = subect to = =, =, = S, =,... = S, + = = Thus, the lear (uder-estate of the optal value fucto v( s α +β, where α = U + V S, =, ( + ( + β= V V CPL/ 5/29/2002 page 7 CPL/ 5/29/2002 page 8
D. L. Bricker, 2002 Dept of Mechanical & Industrial Engineering The University of Iowa. CPL/XD 12/10/2003 page 1
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, =,
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