{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 14 0 0 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0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 262 "" 0 "" {TEXT -1 21 "The Simplex Algorithm" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Define functions:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 79 "E_2 := proc(n, i, alpha)\n mulrow(array(iden tity, 1..n,1..n), i, alpha);\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Pivot:=proc(A,i,j) linalg[pivot](E_2(rowdim(A),i,1/A[ i,j]) &* A,i,j) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&PivotGR6%% \"AG%\"iG%\"jG6\"F*F*-&%'linalgG6#%&pivotG6%-%#&*G6$-%$E_2G6%-%'rowdim G6#9$9%*&\"\"\"F=&F:6$F;9&!\"\"F:F;F@F*F*F*" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 267 11 "Example of " }{TEXT -1 0 "" }{TEXT 282 20 " Linear Ine qualities" }{TEXT -1 1 ":" }{TEXT 284 1 " " }{TEXT -1 21 " Consider t he system" }}{PARA 0 "" 0 "" {TEXT -1 35 " \+ " }{TEXT 260 25 " x + y <= 1" }}{PARA 260 "" 0 " " {TEXT -1 59 " 2 x + y <= 2" }}{PARA 261 "" 0 "" {TEXT -1 61 " \+ x , y >= 0" }}{PARA 0 "" 0 "" {TEXT -1 33 "written in standard form as: " }{TEXT 261 23 "A v = b, v >= 0 " } {TEXT -1 7 "with " }{TEXT 262 23 "v = [x, y, s1, s2]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A:= matrix(2,4,[1,1,1,0,2,1, 0,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7&\"\"\" F*F*\"\"!7&\"\"#F*F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "b :=vector(2,[1, 2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'vector G6#7$\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Ab:=aug ment(A,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AbG-%'matrixG6#7$7'\" \"\"F*F*\"\"!F*7'\"\"#F*F+F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Pivot(Ab,1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6# 7$7'\"\"\"F(F(\"\"!F(7'F)!\"\"!\"#F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Pivot(%,2,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'m atrixG6#7$7'\"\"\"\"\"!!\"\"F(F(7'F)F(\"\"#F*F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "this is the " }{TEXT 283 4 "BFS " }{TEXT -1 6 " with " }{TEXT 285 15 "basic variables" }{TEXT -1 3 " " }{TEXT 263 16 "x \+ = 1, y = 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Pivot(Piv ot(Ab,1,1),2,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7'\" \"\"#F(\"\"#\"\"!F)F(7'F+F)F(#!\"\"F*F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "this is the " }{TEXT 286 4 "BFS " }{TEXT -1 5 "with " } {TEXT 287 15 "basic variables" }{TEXT -1 2 " " }{TEXT 264 17 "x = 1, s1 = 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Pivot(Pivot(A b,1,1),2,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7'\"\"\" F(F(\"\"!F(7'F)!\"\"!\"#F(F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "t his is the " }{TEXT 288 4 "BFS " }{TEXT -1 5 "with " }{TEXT 289 16 "ba sic variables " }{TEXT -1 1 " " }{TEXT 265 18 " x = 1, s2 = 0" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The above three BFS's represent th e " }{TEXT 290 11 "same vertex" }{TEXT -1 9 ", namely " }{TEXT 266 17 " x = 0, y = 1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 18 "Example of Simplex" }}{PARA 0 "" 0 "" {TEXT -1 20 " " }{TEXT 271 20 " max 3 x + 2 y" }}{PARA 0 "" 0 "" {TEXT -1 21 " \+ " }{TEXT 272 6 " s.t. " }{TEXT -1 2 " " }{TEXT 256 3 " \+ " }{TEXT 270 22 " x + y <= 1" }}{PARA 256 "" 0 "" {TEXT -1 56 " 2 x + y <= 2" }}{PARA 257 "" 0 "" {TEXT -1 58 " x , \+ y >= 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A:=matrix( 2,4,[1,1,1,0,2,1,0,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'ma trixG6#7$7&\"\"\"F*F*\"\"!7&\"\"#F*F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "c:=vector(4,[3,2,0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG-%'vectorG6#7&\"\"$\"\"#\"\"!F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Ac:=stackmatrix(-c,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AcG-%'matrixG6#7%7&!\"$!\"#\"\"!F,7&\"\"\"F.F.F,7&\" \"#F.F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "b:=vector(3,[0 ,1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'vectorG6#7%\"\"!\" \"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Abc:=augment(A c,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AbcG-%'matrixG6#7%7'!\"$! \"#\"\"!F,F,7'\"\"\"F.F.F,F.7'\"\"#F.F,F.F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Pivot(Abc,2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%'matrixG6#7%7'\"\"!\"\"\"\"\"$F(F*7'F)F)F)F(F)7'F(!\"\"!\"#F)F(" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "an " }{TEXT 273 17 " optimal solut ion" }{TEXT -1 7 " with " }{TEXT 274 36 " basic variables x = 1, s 2 = 0 " }{TEXT -1 7 " and " }{TEXT 275 18 "optimal value = 3" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Another path taken by the algorith m:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Pivot(Abc,2,2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7'!\"\"\"\"!\"\"#F)F*7' \"\"\"F,F,F)F,7'F,F)F(F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Pivot(%,3,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7' \"\"!F(\"\"\"F)\"\"$7'F(F)\"\"#!\"\"F(7'F)F(F-F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "an " }{TEXT 277 17 "optimal solution " }{TEXT -1 7 " with " }{TEXT 278 15 "basic variables" }{TEXT -1 2 " " }{TEXT 276 13 "x = 1, y = 0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 36 "The Big M Method for computing a BFS" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "A:=matrix(3,6,[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,19]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7(\"\"\"\"\"#\" \"$\"\"%\"\"&\"\"'7(\"\"(\"\")\"\"*\"#5\"#6\"#77(\"#8\"#9\"#:\"#;\"#< \"#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b:=vector(3,[21,57, 94]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'vectorG6#7%\"#@\"#d \"#%*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A:=augment(A,array (identity,1..3,1..3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'mat rixG6#7%7+\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'F*\"\"!F07+\"\"(\"\")\"\"*\"# 5\"#6\"#7F0F*F07+\"#8\"#9\"#:\"#;\"#<\"#>F0F0F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "Ab:=augment(A,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AbG-%'matrixG6#7%7,\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'F* \"\"!F0\"#@7,\"\"(\"\")\"\"*\"#5\"#6\"#7F0F*F0\"#d7,\"#8\"#9\"#:\"#;\" #<\"#>F0F0F*\"#%*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "c:=vec tor(10,[0,0,0,0,0,0,M,M,M,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" cG-%'vectorG6#7,\"\"!F)F)F)F)F)%\"MGF*F*F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "Abc:=stackmatrix(c,Ab);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AbcG-%'matrixG6#7&7,\"\"!F*F*F*F*F*%\"MGF+F+F*7,\"\" \"\"\"#\"\"$\"\"%\"\"&\"\"'F-F*F*\"#@7,\"\"(\"\")\"\"*\"#5\"#6\"#7F*F- F*\"#d7,\"#8\"#9\"#:\"#;\"#<\"#>F*F*F-\"#%*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "rowdim(Abc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "for k from 1 to cold im(Abc) do Abc[1,k]:=Abc[1,k]-M*sum(Abc[i,k],i=2..rowdim(Abc)) od:eval m(Abc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7,,$%\"MG!#@, $F)!#C,$F)!#F,$F)!#I,$F)!#L,$F)!#P\"\"!F5F5,$F)!$s\"7,\"\"\"\"\"#\"\"$ \"\"%\"\"&\"\"'F9F5F5\"#@7,\"\"(\"\")\"\"*\"#5\"#6\"#7F5F9F5\"#d7,\"#8 \"#9\"#:\"#;\"#<\"#>F5F5F9\"#%*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Pivot(%,4,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7 &7,\"\"!,$%\"MG#!#=\"#8,$F*#!#OF-,$F*#!#aF-,$F*#!#sF-,$F*#!##)F-F(F(,$ F*#\"#@F-,$F*#!$i#F-7,F(#\"#7F-#\"#CF-#\"#OF-#\"#[F-#\"#fF-\"\"\"F(#! \"\"F-#\"$z\"F-7,F(#\"\"'F-FA#\"#=F-FC#\"#BF-F(FK#!\"(F-#\"#$)F-7,FK# \"#9F-#\"#:F-#\"#;F-#\"#F-F(F(#FKF-#\"#%*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Pivot(%,3,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7,\"\"!F(F(F(F(,$%\"MG!\"\"F(,$F*\"\"$F(F )7,F(F(F(F(F(\"\"\"F/!\"#F/F/7,F(#F/\"\"%#F/\"\"##F-F3F/#\"#B\"#CF(#\" #8F9#!\"(F9#\"#$)F97,F/F6F4F2F(#\"\"&F9F(#!# " 0 "" {MPLTEXT 1 0 30 "delrows(delcols(%,7..9),1..1 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7)\"\"!F(F(F(F(\" \"\"F)7)F(#F)\"\"%#F)\"\"##\"\"$F,F)F(#\"\"&F.7)F)F/F-F+F(F(F1" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "This is the " }{TEXT 296 35 "BFS \+ x1 = 5/2, x4 = 5/2, x5 = 1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 44 "Example of Simplex with artificial variables" }{TEXT -1 23 " (Tex t, p. 91, Ex. 11)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 " \+ " }{TEXT 257 26 "max 5 x + 2 y + 3 z" }}{PARA 256 " " 0 "" {TEXT -1 66 " s.t. x + y + \+ z = 1" }}{PARA 257 "" 0 "" {TEXT -1 62 " \+ 2 x + 5 y + 3 z <= 4" }}{PARA 258 "" 0 "" {TEXT -1 63 " 4 x + y + 3 z \+ <= 2" }}{PARA 259 "" 0 "" {TEXT -1 68 " \+ x , y , z >= 0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 17 "Phase 1: Find BFS" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A:=matrix(3,6,[1,1,1,1,0,0,2,5,3,0,1,0,4,1,3,0,0,1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7(\"\"\"F*F*F*\"\" !F+7(\"\"#\"\"&\"\"$F+F*F+7(\"\"%F*F/F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "c:=vector(6,[0,0,0,-M,0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG-%'vectorG6#7(\"\"!F)F),$%\"MG!\"\"F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "b:=vector(5,[0,0,1,4,2]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'vectorG6#7'\"\"!F)\"\"\"\"\" %\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Ac:=stackmatrix(- c,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AcG-%'matrixG6#7&7(\"\"!F* F*%\"MGF*F*7(\"\"\"F-F-F-F*F*7(\"\"#\"\"&\"\"$F*F-F*7(\"\"%F-F1F*F*F- " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "c:=vector(6,[5,2,3,0,0, 0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG-%'vectorG6#7(\"\"&\"\"# \"\"$\"\"!F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Ac:=stack matrix(-c,Ac);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AcG-%'matrixG6#7' 7(!\"&!\"#!\"$\"\"!F-F-7(F-F-F-%\"MGF-F-7(\"\"\"F1F1F1F-F-7(\"\"#\"\"& \"\"$F-F1F-7(\"\"%F1F5F-F-F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Abc:=augment(Ac,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AbcG-% 'matrixG6#7'7)!\"&!\"#!\"$\"\"!F-F-F-7)F-F-F-%\"MGF-F-F-7)\"\"\"F1F1F1 F-F-F17)\"\"#\"\"&\"\"$F-F1F-\"\"%7)F6F1F5F-F-F1F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "for j from 1 to coldim(Abc) do" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 45 "Abc[2,j]:=Abc[2,j]-Abc[3,j]*M od: evalm(Abc) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7)!\"&!\"#!\"$\"\"! F+F+F+7),$%\"MG!\"\"F-F-F+F+F+F-7)\"\"\"F1F1F1F+F+F17)\"\"#\"\"&\"\"$F +F1F+\"\"%7)F6F1F5F+F+F1F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Pivot(Abc,4,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7 )#!#@\"\"&\"\"!#!\"*F*F+#\"\"#F*F+#\"\")F*7),$%\"MG#!\"$F*F+,$F4#!\"#F *F+,$F4#\"\"\"F*F+,$F4#!\"\"F*7)#\"\"$F*F+F.FF+F;7)F.FF<#\"\"'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Pivot(%,3,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'m atrixG6#7'7)#!\"$\"\"#\"\"!F+#\"\"*F*#!\"\"F*F+#\"\"&F*7)F+F+F+%\"MGF+ F+F+7)#\"\"$F*F+\"\"\"F0F.F+#F7F*7)F.F7F+F(F8F+F87)F+F+F+!\"'F7F7F+" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 293 4 "BFS:" }{TEXT -1 2 " " }{TEXT 292 65 "y = 1/2 , z = 1/2 , s3 = 0. Drop the artificial varia ble" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "delrows(delcols(%,4. .4),2..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7(#!\"$\" \"#\"\"!F+#!\"\"F*F+#\"\"&F*7(#\"\"$F*F+\"\"\"F,F+#F3F*7(F,F3F+F4F+F47 (F+F+F+F3F3F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 30 "Phase 2: Find o ptimal solution" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Pivot(%, 2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7(\"\"!F(\"\"\" !\"\"F(\"\"$7(F)F(#\"\"#F+#F*F+F(#F)F+7(F(F)F0F0F(F-7(F(F(F(F)F)F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Pivot(%,4,4);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7(\"\"!F(\"\"\"F(F)\"\"$7(F)F(#\" \"#F*F(#F)F*F.7(F(F)F.F(#!\"\"F*F,7(F(F(F(F)F)F(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 258 16 "optimal solution" }{TEXT -1 8 " with " }{TEXT 281 17 "basic variables " }{TEXT -1 2 " " }{TEXT 280 31 "x = 1/3, \+ y = 2/3, s2 = 0" }{TEXT -1 7 ", and " }{TEXT 259 12 "optimal val u" }{TEXT -1 3 "e " }{TEXT 279 4 "= 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "43 4 0" 68 }{VIEWOPTS 1 1 0 1 1 1803 }