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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 29 "Module 3 : Finite Mathematics" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 24 "301 : Lin
ear Programming" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 17 "O B J E C T I V E" }}{PARA 0 "" 0 "" {TEXT -1 156 "In thi
s project we will graph linear inequalities in 2 dimensions and solve \+
linear programming problems of optimizing a function over these planar
 regions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 
-1 9 "S E T U P" }}{PARA 0 "" 0 "" {TEXT -1 252 "In this project we wi
ll use the following command packages. Type and execute this line befo
re begining the project below. If you re-enter the worksheet for this \+
project, be sure to re-execute this statement before jumping to any po
int in the worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}{PARA 7 "" 1 "" 
{TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________
____________________________________________________________________" 
}}{PARA 4 "" 0 "" {TEXT -1 50 "A. Graphing Linear Inequalities & Feasi
ble Regions" }}{PARA 0 "" 0 "" {TEXT -1 83 "__________________________
_________________________________________________________" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 93 "One of the simplest regions we can create is a single i
nequality. The result is a half-plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " sys :=  \{  y >= -.8*x \+
+ 15 \};\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG<#1,&%\"xG$!\")!
\"\"\"#:\"\"\"%\"yG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 " in
equal( sys, x = -5..12,  y = -4..18, optionsfeasible = (color = red),
\n  optionsexcluded = (color = blue), optionsopen    = (color = white)
,  \n  thickness = 1, linestyle = 3, optionsclosed   = (color = coral,
 thickness = 2) );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
{PLOTDATA 2 "6(-%)POLYGONSG6$7'7$$!\"&\"\"!$!\"%F*7$F($\"#=F*7$$!0++++
++v$!#9F.7$$\"#7F*$\"0++++++S&F37$F5F+-%'COLOURG6&%$RGBG$F*F*F>$\"*+++
+\"!\")-F$6$7'F'F-7$F5F.F9F'-F;6&F=F?F>F>-%'CURVESG6%7$F0F4-F;6&F=F?$
\")AR!)\\FAF>-%*THICKNESSG6#\"\"#-FQ6#\"\"\"-%*LINESTYLEG6#\"\"$-%&STY
LEG6#%,PATCHNOGRIDG" 1 3 0 3 10 1 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The intersection of two half-pl
anes is an unbounded triangular region." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " sys :=  \{  y < x + 7,
 x + y <= 16 \};\n    " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG<$2%
\"yG,&%\"xG\"\"\"\"\"(F*1,&F)F*F'F*\"#;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 227 "inequal( sys, x = -5..12,  y = -4..18, optionsfeasib
le = (color = red),\n  optionsexcluded = (color = blue), optionsopen  \+
  = (color = white),  \n  thickness = 1, linestyle = 3, optionsclosed \+
  = (color = coral, thickness = 2) );" }}{PARA 13 "" 1 "" {GLPLOT2D 
400 300 300 {PLOTDATA 2 "6)-%)POLYGONSG6%7%7$$!\"#\"\"!$\"#=F*7$$\"#7F
*F+7$F.$\"\"%F*7%7$$!\"&F*F+7$F.F.7$$!\"%F*F9-%'COLOURG6&%$RGBG$F*F*F?
$\"*++++\"!\")-F$6$7'7$F5F9F4F-7$F.F9FF-F<6&F>F@F?F?-%'CURVESG6%7$F'F0
-F<6&F>F@$\")AR!)\\FBF?-%*THICKNESSG6#\"\"#-FK6$7$F7F8-F<6&F>\"\"\"Fen
Fen-FS6#Fen-%*LINESTYLEG6#\"\"$-%&STYLEG6#%,PATCHNOGRIDG" 1 3 0 3 10 
1 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv
e 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 124 "Completely enclosed regions can also be formed by creati
ng a system of inequalities which bind the region in all directions." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 66 " sys := \{ x + y <= 16, x > -3, x < 10, y > -2, y <= 14   \};\n \+
     " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG<'2!\"$%\"xG2F(\"#52!
\"#%\"yG1F-\"#91,&F(\"\"\"F-F2\"#;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 227 "inequal( sys, x = -5..12,  y = -4..18, optionsfeasib
le = (color = red),\n  optionsexcluded = (color = blue), optionsopen  \+
  = (color = white),  \n  thickness = 1, linestyle = 3, optionsclosed \+
  = (color = coral, thickness = 2) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%)POLY
GONSG6(7&7$$!\"&\"\"!$!\"%F*7$F($\"#=F*7$$F*F*F.7$F1F+7&F'7$F(F17$$\"#
7F*F17$F6F+7&7$F($\"#9F*F-7$F6F.7$F6F;7&F0F=F8F27%7$$!\"#F*F.F=7$F6$\"
\"%F*-%'COLOURG6&%$RGBGF1F1$\"*++++\"!\")-F$6$7'F'F-F=F8F'-FH6&FJFKF1F
1-%'CURVESG6&7$F:F>7$FAFD-FH6&FJFK$\")AR!)\\FMF1-%*THICKNESSG6#\"\"#-F
T6&7$F2F07$F4F5F\\o-FH6&FJ\"\"\"F`oF`o-Fgn6#F`o-%*LINESTYLEG6#\"\"$-%&
STYLEG6#%,PATCHNOGRIDG" 1 3 0 3 10 1 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________
__________________________________________________________" }}{PARA 4 
"" 0 "" {TEXT -1 15 "B. Optimization" }}{PARA 0 "" 0 "" {TEXT -1 83 "_
______________________________________________________________________
____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 924 "In this section, we will solve
 optimization problems in linear programming. A typical problem of thi
s sort has two basic components :\n     the contraints - a system of i
nequalities which define a feasible region. \n     an objective functi
on to maximize or minimize on this region\n\011( The regions defined a
nd graphed above are examples of feasible regions. The one difference \+
is that no \011\n\011inequalities can be strict inequalities. In other
 words, every inequality must be  or .)\n\nThe goal of the problem is \+
to find a point (x,y) where the value of objective function is maximal
 or minimal, and what the value of the objective function is at that p
oint. We are going to break the problem down into 5 steps.\n\0111.  De
fine a System of Inequalities (the constraints)\n\0112.  Graph the Fea
sible Region (an optional but advisable step)\n\0113.  Define the Obje
ctiive Function, P(x,y)\n\0114.  Find the Solution, (x,y)\n\0115.  Fin
d the Optimal Value" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 98 "Lets look at an example.Well take the problem that foll
ows and solve in Maple using these 5 steps." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "1.  Define a System of Inequali
ties " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 " sys :=  \{ x + y <= 20, x >=0, x <= 12, y >= 0, y <=
 14   \};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG<'1,&%\"xG\"\"\"%
\"yGF)\"#?1\"\"!F(1F(\"#71F-F*1F*\"#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 30 "2.  Graph the Feasible Region " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
164 " inequal( sys,   x = -2..15,  y = -2..18, optionsfeasible = (colo
r = red),\n       optionsexcluded = (color = blue), optionsclosed = (c
olor = coral), thickness = 2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
300 {PLOTDATA 2 "6'-%)POLYGONSG6(7%7$$\"\"#\"\"!$\"#=F*7$$\"#:F*F+7$F.
$\"\"&F*7&7$$!\"#F*F57$F5F+7$$F*F*F+7$F9F57&7$$\"#7F*F+F-7$F.F57$F=F57
&F47$F5F97$F.F9F?7&7$F5$\"#9F*F7F-7$F.FF-%'COLOURG6&%$RGBGF9F9$\"*++++
\"!\")-F$6$7'F4F7F-F?F4-FJ6&FLFMF9F9-%'CURVESG6(7$F'F07$F:F87$F@F<7$FC
FB7$FEFH-FJ6&FLFM$\")AR!)\\FOF9-%*THICKNESSG6#F)-%&STYLEG6#%,PATCHNOGR
IDG" 1 3 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 52 "This is very similar to the graphs we created above.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "3. De
fine the Objective Function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 186 "Maple defines functions in this way. Its not e
nough to simply type P(x,y) =  2x + 3y + 11. You need to type P := (x,
y) -> and then the definition of the function followed by a semicolon.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 " P := 2*x + 3*y + 11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"P
G,(%\"xG\"\"#*&\"\"$\"\"\"%\"yGF*F*\"#6F*" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "4. Find the Solution" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Maple is able t
o find the solution automatically once we have defined the system and \+
the function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "with(simplex):" }}{PARA 7 "" 1 "" {TEXT -1 45 "W
arning, the name display has been redefined\n" }}{PARA 7 "" 1 "" 
{TEXT -1 87 "Warning, the protected names maximize and minimize have b
een redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 27 " soln := maximize( P, sys);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%%solnG<$/%\"yG\"#9/%\"xG\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 25 "5. Find the Optimal Value" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Here we substituted \+
the solution (x,y) into the objective function to find the maximal val
ue." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 " subs( soln, P(x,y) );" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,(-\"\"'6$F%\"#9\"\"#*&\"\"$\"\"\"-F'F&F+F+\"#6F+" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "Note that the sol
ution depends on both the region and the function. If we change the fu
nction, we can get a completely different solution and optimal value.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 28 " P := (x,y) ->  7*x + 2*y ;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"PGR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&9$\"\"(*&\"\"#\"\"\"
9%F2F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " soln := ma
ximize( P(x,y), sys);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG<$/
%\"xG\"#7/%\"yG\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " su
bs( soln, P(x,y) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$+\"" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 24 "Here is another problem." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "1.  Define a System of In
equalities (the constraints)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " sys :=  \{ x >= 0, x <= 16,
  y >= 0,  \n           2*y <= x + 20 \}; " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$sysG<&1\"\"!%\"xG1F'%\"yG1F(\"#;1,$F*\"\"#,&F(\"\"\"
\"#?F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
62 "2.  Graph the Feasible Region (an optional but advisable step)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
215 " inequal( sys ,  x = -1..18,  y = -1..20,\n          optionsfeasi
ble = (color = red),\n          optionsexcluded = (color = blue),\n   \+
       optionsclosed   = (color = coral,\n                         thi
ckness = 2) );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "
6&-%)POLYGONSG6'7&7$$!\"\"\"\"!F(7$F($\"#?F*7$$F*F*F,7$F/F(7&F'7$F(F/7
$$\"#=F*F/7$F4F(7&7$$\"#;F*F,7$F4F,F67$F9F(7&7$F($\"0++++++]*!#9F+F;7$
F4$\"#>F*-%'COLOURG6&%$RGBGF/F/$\"*++++\"!\")-F$6$7'F'F+F;F6F'-FF6&FHF
IF/F/-%'CURVESG6(7$F0F.7$F2F37$F<F87$FBF>-FF6&FHFI$\")AR!)\\FKF/-%*THI
CKNESSG6#\"\"#-%&STYLEG6#%,PATCHNOGRIDG" 1 3 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "3.  Defin
e the Objective Function, P(x,y)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " P := (x,y) ->  15*x - 3*y;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGR6$%\"xG%\"yG6\"6$%)operator
G%&arrowGF),&9$\"#:*&\"\"$\"\"\"9%F2!\"\"F)F)F)" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "4.  Find the Solution, (x
,y)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 " soln := minimize( P(x,y) , sys );" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%solnG<$/%\"xG\"\"!/%\"yG\"#5" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "5.  Find the Optimal Valu
e " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 " subs( soln, P(x,y) );" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#!#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 \+
0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }