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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 263 20 "Postoptimal Analysis" }
}}{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 -1 10 "          " }{TEXT 262 4 " max" }{TEXT -1 3 " \{ " }
{TEXT 256 1 "c" }{TEXT -1 3 "^T " }{TEXT 257 1 "x" }{TEXT -1 6 " :  A \+
" }{TEXT 258 1 "x" }{TEXT -1 4 " <= " }{TEXT 259 1 "b" }{TEXT -1 2 ", \+
" }{TEXT 260 1 "x" }{TEXT -1 3 " >=" }{TEXT 261 2 " 0" }{TEXT -1 7 " \+
\} with" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "A:=matrix(2,3,[3
,4,1,1,-2,3]);b:=vector(2,[8,12]);c:=vector(3,[2,-1,2]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7%\"\"$\"\"%\"\"\"7%F,!\"#F*
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'vectorG6#7$\"\")\"#7" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG-%'vectorG6#7%\"\"#!\"\"F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The original table, with RHS & sla
cks in left, is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "Abc:=a
ugment(vector(3,[0,b[1],b[2]]),augment(array(1..3,1..3,identity),stack
matrix(vector(3,[-c[1],-c[2],-c[3]]),A)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$AbcG-%'matrixG6#7%7)\"\"!\"\"\"F*F*!\"#F+F,7)\"\")F*
F+F*\"\"$\"\"%F+7)\"#7F*F*F+F+F,F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "Pivot(%,2,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'m
atrixG6#7%7)#\"#;\"\"$\"\"\"#\"\"#F*\"\"!F.#\"#6F*#!\"%F*7)#\"\")F*F.#
F+F*F.F+#\"\"%F*F67)#\"#GF*F.#!\"\"F*F+F.#!#5F*F4" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "T:=Pivot(%,3,7);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"TG-%'matrixG6#7%7)\"#5\"\"\"#F+\"\"#F,\"\"!F-F.7)#
\"\"$F-F.#F1\"\")#!\"\"F3F+#\"\"(\"\"%F.7)#F7F-F.F4F2F.#!\"&F8F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The basis M is made of " }{TEXT 
266 15 "columns 2, 5, 7" }{TEXT -1 26 " of Abc (corresponding to " }
{TEXT 265 25 "basic variables z, x3, x5" }{TEXT -1 1 ")" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "M:=augment(submatrix(Abc,1..3,2..2)
,submatrix(Abc,1..3,5..5),submatrix(Abc,1..3,7..7));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7%7%\"\"\"!\"#F+7%\"\"!\"\"$F*7%F-
F*F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "The inverse of the basis,
 Minv,  is where the slack variables used to be" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "Minv:=submatrix(T,1..3,2..4);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%MinvG-%'matrixG6#7%7%\"\"\"#F*\"\"#F+7%\"\"!#\"
\"$\"\")#!\"\"F17%F.F2F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Check
 that Minv = inverse(M)!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 18 "Chan
ging the RHS. " }{TEXT -1 8 "The new " }{TEXT 268 2 "b " }{TEXT -1 2 "
is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "bnew:=vector(2,[15,4]
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%bnewG-%'vectorG6#7$\"#:\"\"%
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Column 1 is changed to  refle
ct the new RHS. The new column 1 is" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "evalm(Minv &* vector(3,[0,bnew[1],bnew[2]]));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%#\"#>\"\"##\"#T\"\")#!\"
$F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "The new table is" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Tnew:=augment(%,submatrix(T,
1..3,2..7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%TnewG-%'matrixG6#7%
7)#\"#>\"\"#\"\"\"#F-F,F.\"\"!F,F/7)#\"#T\"\")F/#\"\"$F3#!\"\"F3F-#\"
\"(\"\"%F/7)#!\"$F3F/F6F4F/#!\"&F:F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 14 "This table is " }{TEXT 269 15 "dually feasible" }{TEXT -1 5 " b
ut " }{TEXT 270 19 "primally infeasible" }{TEXT -1 10 ". Use the " }
{TEXT 272 11 "dual method" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "Pivot(%,3,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7)#
\"#*)\"#5\"\"\"#\"\"$F*#\"#6F*\"\"!F0#\"\")\"\"&7)#\"#BF3F0#F+F3#\"\"#
F3F+F0#\"\"(F37)F,F0#F+F*#!\"$F*F0F+#!\"%F3" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 32 "An optimal table for the new RHS" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 264 19 "Adding a constraint" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 84 "Suppose the constraint    2 x3 - 3 x4 + 6 x5 <= 10 is add
ed to the original problem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "lconstr:=vector(3,[2,-3,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
(lconstrG-%'vectorG6#7%\"\"#!\"$\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "rconstr:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(rc
onstrG\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The basic coefficie
nts of the added cosntraint" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "lconstrB:=vector(3,[0,2,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
)lconstrBG-%'vectorG6#7%\"\"!\"\"#\"\"'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 40 "The inverse of the new basis is computed" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "newMinv:=augment(stackmatrix(Minv,v
ector(3,[0,0,0])),vector(4,[0,0,0,1]));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%(newMinvG-%'matrixG6#7&7&\"\"\"#F*\"\"#F+\"\"!7&F-#\"\"$\"\")#
!\"\"F1F-7&F-F2F/F-7&F-F-F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 91 "for k from 1 to 3 do newMinv[4,k]:=-add(lconstrB[j]*newMinv[j,k]
,j=1..3) od:evalm(newMinv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matr
ixG6#7&7&\"\"\"#F(\"\"#F)\"\"!7&F+#\"\"$\"\")#!\"\"F/F+7&F+F0F-F+7&F+F
+!\"#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The new problem data" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "newAbc:=augment(submatrix
(Abc,1..3,1..4),vector(3,[0,0,0]),submatrix(Abc,1..3,5..7));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'newAbcG-%'matrixG6#7%7*\"\"!\"\"\"F*F*F*!
\"#F+F,7*\"\")F*F+F*F*\"\"$\"\"%F+7*\"#7F*F*F+F*F+F,F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "newAbc:=evalm(stackmatrix(newAbc,ve
ctor(8,[rconstr,0,0,0,1,2,-3,6])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%'newAbcG-%'matrixG6#7&7*\"\"!\"\"\"F*F*F*!\"#F+F,7*\"\")F*F+F*F*\"
\"$\"\"%F+7*\"#7F*F*F+F*F+F,F/7*\"#5F*F*F*F+\"\"#!\"$\"\"'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 275 9 "new table" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "newT:=evalm(newMinv &* newAbc);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%newTG-%'matrixG6#7&7*\"#5\"\"\"#F+
\"\"#F,\"\"!F.F-F.7*#\"\"$F-F.#F1\"\")#!\"\"F3F.F+#\"\"(\"\"%F.7*#F7F-
F.F4F2F.F.#!\"&F8F+7*!#9F.F.!\"#F+F.F+F." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 271 15 "Dually feasible" }{TEXT -1 2 ", " }{TEXT 273 19 "primall
y infeasible" }{TEXT -1 5 ". Do " }{TEXT 274 15 "dual iterations" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Pivot(%,4,4);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7*#\"#8\"\"#\"\"\"#F+F*\"\"!#F+\"
\"%F-#\"\"*F/F-7*#\"#>\"\")F-#\"\"$F5F-#!\"\"\"#;F+#\"#FF:F-7*#\"\"(F5
F-#F9F5F-#F7F:F-#!#<F:F+7*F?F-F-F+#F9F*F-FEF-" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 19 "An optimal solution" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
283 16 "A bigger example" }{TEXT 285 1 ":" }{TEXT -1 41 "    Text, p. \+
121, no. 20:                " }{TEXT 282 4 " max" }{TEXT -1 3 " \{ " }
{TEXT 276 1 "c" }{TEXT -1 3 "^T " }{TEXT 277 1 "x" }{TEXT -1 6 " :  A \+
" }{TEXT 278 1 "x" }{TEXT -1 4 " <= " }{TEXT 279 1 "b" }{TEXT -1 2 ", \+
" }{TEXT 280 1 "x" }{TEXT -1 3 " >=" }{TEXT 281 2 " 0" }{TEXT -1 7 " \+
\} with" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "A:=matrix(3,4,[2
,1,3,1,2,3,0,4,3,1,2,0]);b:=vector(3,[8,12,18]);c:=vector(4,[1,2,1,1])
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7&\"\"#\"\"\"
\"\"$F+7&F*F,\"\"!\"\"%7&F,F+F*F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"bG-%'vectorG6#7%\"\")\"#7\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"cG-%'vectorG6#7&\"\"\"\"\"#F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
50 "The original table, with RHS & slacks in left, is:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "Abc:=matrix([[0, 1, 0, 0, 0, -1, -
2, -1, -1], [8, 0, 1, 0, 0, 2, 1, 3, 1], [12, 0, 0, 1, 0, 2, 3, 0, 4],
 [18, 0, 0, 0, 1, 3, 1, 2, 0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
$AbcG-%'matrixG6#7&7+\"\"!\"\"\"F*F*F*!\"\"!\"#F,F,7+\"\")F*F+F*F*\"\"
#F+\"\"$F+7+\"#7F*F*F+F*F0F1F*\"\"%7+\"#=F*F*F*F+F1F+F0F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 27 "Consider the optimal table," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "T:=matrix([[28/3, 1, 1/3, 5/9, 0, \+
7/9, 0, 0, 14/9], [4/3, 0, 1/3, -1/9, 0, 4/9, 0, 1, -1/9], [4, 0, 0, 1
/3, 0, 2/3, 1, 0, 4/3], [34/3, 0, -2/3, -1/9, 1, 13/9, 0, 0, -10/9]]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG-%'matrixG6#7&7+#\"#G\"\"$\"
\"\"#F-F,#\"\"&\"\"*\"\"!#\"\"(F1F2F2#\"#9F17+#\"\"%F,F2F.#!\"\"F1F2#F
9F1F2F-F:7+F9F2F2F.F2#\"\"#F,F-F2F87+#\"#MF,F2#!\"#F,F:F-#\"#8F1F2F2#!
#5F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The basis M is made of " 
}{TEXT 287 18 "columns 2, 8, 7, 5" }{TEXT -1 26 " of Abc (correspondin
g to " }{TEXT 286 30 "basic variables x1, x7, x6, x4" }{TEXT -1 1 ")" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "M:=augment(submatrix(Abc
,1..4,2..2),submatrix(Abc,1..4,8..8),submatrix(Abc,1..4,7..7),submatri
x(Abc,1..4,5..5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG
6#7&7&\"\"\"!\"\"!\"#\"\"!7&F-\"\"$F*F-7&F-F-F/F-7&F-\"\"#F*F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "The inverse of the basis, Minv,  i
s where the slack variables used to be" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "Minv:=submatrix(T,1..4,2..5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%MinvG-%'matrixG6#7&7&\"\"\"#F*\"\"$#\"\"&\"\"*\"\"!7
&F0F+#!\"\"F/F07&F0F0F+F07&F0#!\"#F,F2F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Check that Minv =
 inverse(M)!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 19 "Adding a constra
int" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Suppose the constraint    \+
2 x5 + 3 x6 - x7 + 12 x8 <= 10 is added to the problem." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "lconstr:=vector(4,[2,3,-1,12]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lconstrG-%'vectorG6#7&\"\"#\"\"$!\"
\"\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "rconstr:=10;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(rconstrG\"#5" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 46 "The basic coefficients of the added cosntraint" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "lconstrB:=vector(4,[0,-1,3,0
]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)lconstrBG-%'vectorG6#7&\"\"!
!\"\"\"\"$F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "newMinv:=au
gment(stackmatrix(Minv,vector(4,[0,0,0,0])),vector(5,[0,0,0,0,1]));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(newMinvG-%'matrixG6#7'7'\"\"\"#F*
\"\"$#\"\"&\"\"*\"\"!F07'F0F+#!\"\"F/F0F07'F0F0F+F0F07'F0#!\"#F,F2F*F0
7'F0F0F0F0F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "for k from \+
1 to 4 do newMinv[5,k]:=-add(lconstrB[j]*newMinv[j,k],j=1..4) od:evalm
(newMinv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'\"\"\"#F
(\"\"$#\"\"&\"\"*\"\"!F.7'F.F)#!\"\"F-F.F.7'F.F.F)F.F.7'F.#!\"#F*F0F(F
.7'F.F)#!#5F-F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "newAbc
:=augment(submatrix(Abc,1..4,1..5),vector(4,[0,0,0,0]),submatrix(Abc,1
..4,6..9));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'newAbcG-%'matrixG6#7
&7,\"\"!\"\"\"F*F*F*F*!\"\"!\"#F,F,7,\"\")F*F+F*F*F*\"\"#F+\"\"$F+7,\"
#7F*F*F+F*F*F0F1F*\"\"%7,\"#=F*F*F*F+F*F1F+F0F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 76 "newAbc:=evalm(stackmatrix(newAbc,vector(10,[rc
onstr,0,0,0,0,1,2,3,-1,12])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'n
ewAbcG-%'matrixG6#7'7,\"\"!\"\"\"F*F*F*F*!\"\"!\"#F,F,7,\"\")F*F+F*F*F
*\"\"#F+\"\"$F+7,\"#7F*F*F+F*F*F0F1F*\"\"%7,\"#=F*F*F*F+F*F1F+F0F*7,\"
#5F*F*F*F*F+F0F1F,F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "new
T:=evalm(newMinv &* newAbc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%new
TG-%'matrixG6#7'7,#\"#G\"\"$\"\"\"#F-F,#\"\"&\"\"*\"\"!F2#\"\"(F1F2F2#
\"#9F17,#\"\"%F,F2F.#!\"\"F1F2F2#F9F1F2F-F:7,F9F2F2F.F2F2#\"\"#F,F-F2F
87,#\"#MF,F2#!\"#F,F:F-F2#\"#8F1F2F2#!#5F17,FCF2F.FGF2F-F<F2F2#\"#rF1
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Dually feasible, primally inf
easible. Do dual iterations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "Pivot(%,5,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7,
\"\"*\"\"\"#F)\"\"#\"\"!F,F*F)F,F,#\"#6F+7,#\"\"(\"\"&F,#\"\"$\"#5F,F,
#!\"\"F5#F+F2F,F)#!\"*F57,#\"#>F2F,#F)F5F,F,F3#\"\"%F2F)F,#\"#PF57,#\"
#dF2F,#!\"(F5F,F)F6F0F,F,#!#>F57,#F4F2F,#!\"$F5F)F,F9#!\"#F2F,F,#!#rF5
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "An optimal solution" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "68 0 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 }