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{TEXT 256 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 130 "Minimum cost flow proble ms can be set up as a linear programming problems, although the overh ead gets out of hand rather quickly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "For example, the problem in our text \+ which is used to explicate the network simplex algorithm is easy to ch eck using simplex." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "resta rt;with(linalg):with(networks):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warnin g, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, ne w definition for trace" }}{PARA 7 "" 1 "" {TEXT -1 36 "Warning, new de finition for charpoly" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new def inition for rank" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "new(G): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "addvertex([A,B,C,D,E],w eights=[40,50,0,-30,-60],G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%\"CG% \"AG%\"EG%\"DG%\"BG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "add edge([[A,C],[A,D],[A,B],[B,C],[C,E],[E,D],[D,E]],names=[xAC,xAD,xAB,xB C,xCE,xED,xDE],\nweights=[4,9, 2,3,1,2,3],G):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "draw(G);" }}{PARA 13 "" 1 "" {INLPLOT "64-%'CURVE SG6$7$7$$\"+Q*p,4$!#5$\"+l^c5&*F*7$$!+]*p,4)F*$\"+9D&y(eF*-%'COLOURG6& %$RGBG\"\"!$\"#5!\"\"F6-F$6$7$7$$\"\"\"F6F67$$!+M*p,4)F*$!+PD&y(eF*F2- F$6$7$F=F'F2-F$6$7$F@7$$\"+l*p,4$F*$!+c^c5&*F*F2-F$6$7$FKF@F2-F$6$7$F- FKF2-%'POINTSG6#F'-%%TEXTG6$F'%\"BG-FW6#F--FZ6$F-%\"CG-FW6#F@-FZ6$F@% \"DG-FW6#FK-FZ6$FK%\"EG-F$6$7$F=F-F2-FW6#F=-FZ6$F=%\"AG-%*AXESSTYLEG6# %%NONEG" 2 211 211 211 2 0 1 0 2 9 0 1 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2480 16495 0 0 0 0 0 0 }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "cost:= sum(eweight(G)[edge s(G)['i']]*edges(G)['i'],'i'=1..nops(edges(G))) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%costG,0%$xACG\"\"%%$xABG\"\"#%$xDEG\"\"$%$xEDGF)%$ xCEG\"\"\"%$xBCGF+%$xADG\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "cost := 4*xAC+3*xDE+2*xED+xCE+3*xBC+2*xAB+9*xAD;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%costG,0%$xACG\"\"%%$xABG\"\"#%$xDEG\"\"$%$xEDGF )%$xCEG\"\"\"%$xBCGF+%$xADG\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "cnsts := [xAB+xAC +xAD=50,-xAB+xBC=40,xCE-xAC-xBC=0, \nxCE+xDE-xED=60,xAD+xED-xDE=30,xCE<=80,xAB<=10]; " }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%&cnstsG7)/,(%$xABG\"\"\"%$xACGF)%$xADGF)\"#]/,&F(! \"\"%$xBCGF)\"#S/,(%$xCEGF)F*F/F0F/\"\"!/,(F4F)%$xDEGF)%$xEDGF/\"#g/,( F+F)F9F)F8F/\"#I1F4\"#!)1F(\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "soln := simplex[minimize](cost,cnsts,NONNEGATIVE);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solnG<)/%$xABG\"\"!/%$xDEGF(/%$xBCG\"#S/% $xEDG\"#?/%$xACGF-/%$xADG\"#5/%$xCEG\"#!)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "subs(soln,cost);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"$!\\" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "So simplex returns the same solution that the network simplex algorithm returns (see page 38 7). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 190 "Let's make up our own five node min cost flow problem. We will defin e it in terms of a tree, then use that to generate a cost function and set of constraints to feed to minimize in simplex." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "new(G):" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 201 "The graph is called G. It has \+ 5 nodes called 1,2,3,4,5; 1 and 2 are source nodes of weights 10 and \+ 20, 3 is a transhipment node (of weight 0) and 4 and 5 are destination nodes of weights 15 and 15. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "addvertex([1,2,3,4,5],\nweights=[10,20,0,-15,-15],G);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"\"\"#\"\"$\"\"%\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Now we'll add the roads (edges). There ar e 8 of them. The edge-weights are the costs. " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 154 "addedge([[1,3],[1,4],[2,3],[2,5],[3,4],[3,5 ],[4,5],[5,4]], names=[x[1,3],x[1,4],x[2,3],x[2,5],x[3,4], x[3,5],x[4, 5],x[5,4]],\nweights=[2,1,1,2,2,3,2,1],G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*&%\"xG6$\"\"\"\"\"$&F$6$F&\"\"%&F$6$\"\"#F'&F$6$F-\"\"& &F$6$F'F*&F$6$F'F0&F$6$F*F0&F$6$F0F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The cost function is generated thusly:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "cost:= sum(eweight (G)[edges(G)[i]]*edges(G)[i], i=1..nops(edges(G)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%costG,2&% \"xG6$\"\"\"\"\"$\"\"#&F'6$F)\"\"%F)&F'6$F+F*F)&F'6$F+\"\"&F+&F'6$F*F. F+&F'6$F*F3F*&F'6$F.F3F+&F'6$F3F.F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 196 "The constraints are 5 in number: one for each node. In addi tion, we have decided there are finite capacities of 5 and 3 on the ed ges [1,4] and [4,5]. This gives two 'upper bound' constraints." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "cnsts := [x[1,3]+x[1,4]=10, x[2,3]+x[2,5]=20,\nx[1,3]+x[2,3]-x[3,4]-x[3,5]=0,\nx[1,4]+x[3,4]+x[5,4 ]-x[4,5]=15,\nx[2,5]+x[3,5]+x[4,5]-x[5,4]=15,x[1,4]<=5,\nx[4,5]<=3];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&cnstsG7)/,&&%\"xG6$\"\"\"\"\"$F+& F)6$F+\"\"%F+\"#5/,&&F)6$\"\"#F,F+&F)6$F5\"\"&F+\"#?/,*F(F+F3F+&F)6$F, F/!\"\"&F)6$F,F8F>\"\"!/,*F-F+F\"#:/,*F6F+F?F +FFF+FDF>FH1F-F81FFF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Now feed. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "soln :=simplex[minimize ](cost,cnsts,NONNEGATIVE);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "`min \+ cost = `,subs(soln,cost);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solnG<*/&%\"xG6$\"\"$\"\"&\"\"!/&F( 6$\"\"%F+F,/&F(6$F+F0F,/&F(6$\"\"#F*F+/&F(6$F7F+\"#:/&F(6$\"\"\"F*F+/& F(6$F*F0\"#5/&F(6$F?F0F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,min~cost ~=~G\"#q" }}}{EXCHG {PARA 285 "" 0 "" {TEXT 257 45 "Generating random \+ minimum cost flow problems." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "We want to define some words to work with minimum cost flow problems." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Here is \+ a word to generate a random set of n positive integers whose sum is s ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "randsupply := proc(s, n)\n local i,j,ra,f,g,l;\nf := rand(n);\n l := vector([seq(1,i=1..n)]) ;\nwhile sum(l[j],j=1..n) " 0 "" {MPLTEXT 1 0 20 "convert(l,list) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " randsupply (1000,4) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"$r\"\"$q(\"\")\"#^" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 177 "Here is a word to set up a minimum cost flow problem w ith m supply nodes with a total supply of su, r transhipment nodes, an d n destination nodes. All capacities are infinite." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1499 "setupmcf := proc(m,su ,r,n,`G`)\n local sup, des, g, sources, destin, trans, i, t, j,k,cost,cnsts, s, d ;\n sup := randsupply(su,m);\n des := randsupply(su,n);\n g := rand(10 0);\n sources := [seq(s.i,i=1..m)];\ndestin := [seq(d.i,i=1..n)];\n tr ans := [seq(t.i,i=1..r)];\n new(G);\n for i from 1 to nops(sources) d o addvertex(sources[i],weights=sup[i],G) od;\nfor t in trans do\naddve rtex(t,weights=0 ,G) od; \nfor j from 1 to nops( destin) do\n addverte x(destin[j],weights= des[j],G) od;\n for s in sources do for d in dest in do\n addedge([s,d],names=x[s,d],weights=g(),G) od od;\nfor s in sou rces do for t in trans do\n addedge([s,t],names=x[s,t],weights=g(),G) \+ od od;\nfor t in trans do for d in destin do\n addedge([t,d],names=x[t ,d],weights=g(),G) od od;\n print( draw(Linear([seq(s.i,i=1..m)],[seq( t.j,j=1..r)],\n[seq(d.k,k=1..n)]),G)) ;\ni := 'i':\ncost := sum(eweigh t(G)[edges(G)[i]]*edges(G)[i],\ni=1..nops(edges(G))); \ncnsts := NULL: i := 'i': j := 'j': k :='k':\nprintlevel:=1;\nfor i from 1 to m do \+ \ncnsts := cnsts,\nconvert([seq( \nop(edges([s.i,d.j],G)),j=1..n)] ,` +`)\n+ convert([ seq(\nop(edges([s.i,t.k],G)),k=1..r)] ,`+`) =\nvweigh t(G)[s.i] od;\nfor i from 1 to r do \ncnsts := cnsts,\n- (convert ([seq( \nop(edges([t.i,d.j],G)),j=1..n)] ,`+`))\n+ convert([ seq(\nop( edges([s.k,t.i],G)),k=1..m)] ,`+`) =\n0 od;\nfor i from 1 to n do \+ \ncnsts := cnsts,\nconvert([seq( \nop(edges([s.j,d.i],G)),j=1..m)] ,` +`)\n+ convert([ seq(\nop(edges([t.k,d.i],G)),k=1..r)] ,`+`) =\nvweigh t(G)[d.i] od;\n \n[cost,[cnsts]]; \nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "We can test this out." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "prob:= setupmcf(2,100,2,3,G);\nsoln:=simplex[minimiz e](prob[1],prob[2],NONNEGATIVE);\nsubs(soln,prob[1]);" }}{PARA 13 "" 1 "" {INLPLOT "6A-%'POINTSG6#7$\"\"\"#F'\"\"#-%%TEXTG6$7$$\"\"*!\"\"F( %#s1G-F$6#7$F'#\"\"$F)-F+6$7$F.F5%#s2G-F$6#7$F)F(-F+6$7$$\"#>F0F(%#t1G -F$6#7$F)F5-F+6$7$FAF5%#t2G-F$6#7$F6\"\"!-F+6$7$$\"$3$!\"#FN%#d1G-F$6# 7$F6F'-F+6$7$FRF'%#d2G-F$6#7$F6F)-F+6$7$FRF)%#d3G-%'CURVESG6$7$F4FX-%' COLOURG6&%$RGBGFN$\"#5F0FN-F_o6$7$FFFinFbo-F_o6$7$F&FMFbo-F_o6$7$FFFXF bo-F_o6$7$F4FFFbo-F_o6$7$F=FMFbo-F_o6$7$F=FXFbo-F_o6$7$F=FinFbo-F_o6$7 $FFFMFbo-F_o6$7$F&F=Fbo-F_o6$7$F&FFFbo-F_o6$7$F4FinFbo-F_o6$7$F&FinFbo -F_o6$7$F4FMFbo-F_o6$7$F4F=Fbo-F_o6$7$F&FXFbo-%*AXESSTYLEG6#%%NONEG" 2 211 211 211 2 0 1 0 2 9 0 1 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 30936 16495 0 0 0 0 0 0 }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%probG7$,B&%\"xG6$%#s2G%#d2G\"#l&F(6$%#t2G %#d3G\"#N&F(6$%#s1G%#d1G\"#Q&F(6$F/F+\"#\\&F(6$F*F/\"#g&F(6$%#t1GF5\"# #)&F(6$F?F+\"##*&F(6$F?F0\"#8&F(6$F/F5\"#x&F(6$F4F?\"#&*&F(6$F4F/\"#u& F(6$F*F0\"#\"*&F(6$F4F0\"#?&F(6$F*F5\"#W&F(6$F*F?\"\"*&F(6$F4F+FF7)/,, F2\"\"\"FfnF[oFSF[oFJF[oFMF[o\"#)*/,,FVF[oF'F[oFPF[oFYF[oF:F[o\"\"#/,, F=!\"\"FAFboFDFboFJF[oFYF[o\"\"!/,,FGFboF7FboF-FboFMF[oF:F[oFco/,*F2F[ oFVF[oF=F[oFGF[oFen/,*FfnF[oF'F[oFAF[oF7F[oFI/,*FSF[oFPF[oFDF[oF-F[o\" #9" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solnG<2/&%\"xG6$%#t2G%#d2G\" \"!/&F(6$%#s1G%#t1GF,/&F(6$F1%#d3G\"\"#/&F(6$F0F+\"#x/&F(6$F0F5\"#7/&F (6$%#s2GF1F6/&F(6$FBF*F,/&F(6$F1F+F,/&F(6$F0F*F,/&F(6$FBF5F,/&F(6$FB%# d1GF,/&F(6$F0FR\"\"*/&F(6$F*FRF,/&F(6$FBF+F,/&F(6$F*F5F,/&F(6$F1FRF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 36 0 0" 0 } {VIEWOPTS 1 1 1 1 1 1803 }