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Cell[TextData[{
  StyleBox["Diet Problem",
    FontColor->RGBColor[0.0557107, 0.137819, 0.517113]],
  "\nAn Application of Vertex Enumeration\nwith ",
  StyleBox["MathLink",
    FontSize->24,
    FontSlant->"Italic",
    FontColor->RGBColor[0.0146487, 0.461387, 0.0967727]],
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Cell[TextData[StyleBox["Komei Fukuda, fukuda@ifor.math.ethz.ch\nSwiss Federal \
Institute of Technology, Lausanne and Zurich\nMarch 14, 1999",
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  FontSlant->"Italic"]], "Subtitle",
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 In this example, the name of the directory is ",
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}], "Text",
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Cell[CellGroupData[{

Cell["What is Diet Problem?", "Section",
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Cell["\<\
The following diet problem is taken from V. Chvatal's  great book \
on Linear Programming (\"Linear Programming\", W.H.Freeman and Company,1983). \
  It is to design a cheapest meal with six possible items below to satisfy \
prescribed nutritional needs.  Please see Page 3 of the book.\
\>", "Text",
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Cell["\<\
var={\"\",\"Oatmeal\",\"Chicken\",\"Eggs\",\"Milk\",\"Cherry Pie\", \

\t\"Pork Beans\"};

price={\"Price/Ser\", \"3c\", \"24c\", \"13c\", \"9c\", \"20c\", \
\"19c\"}\
\>", "Input",
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MatrixForm[dietproblem1=
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 {0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0}, 
 {0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 1}, 
 {4, -1, 0, 0, 0, 0, 0}, {3, 0, -1, 0, 0, 0, 0},
 {2, 0, 0, -1, 0, 0, 0}, {8, 0, 0, 0, -1, 0, 0}, 
 {2, 0, 0, 0, 0, -1, 0}, {2, 0, 0, 0, 0, 0, -1},
 {-2000, 110, 205, 160, 160, 420, 260}, 
 {-55, 4, 32, 13, 8, 4, 14}, 
 {-800, 2, 12, 54, 285, 22, 80}}];\
\>", "Input",
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          {"\<\"\"\>", "\<\"Oatmeal\"\>", "\<\"Chicken\"\>", "\<\"Eggs\"\>", \
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Cell["\<\
m=Transpose[Drop[Transpose[dietproblem1],1]];
b=-First[Transpose[dietproblem1]];
c={3, 24, 13, 9, 20, 19};\
\>", "Input",
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Cell["\<\
We can see the optimal solution better in the following table.   It \
is certainly not an exciting menu.   In fact, an optimal solution to any \
optimization problem tends to be extreme, and thus it must be modified for \
practical purposes.\
\>", "Text",
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Now we try to do something more reasonable.  We use cddmathlink \
fuction AllVertices:\
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(ray) has the first component 1 (0).  If the convex polyhedron is nonempty \
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Cell["\<\
We can then compute ALL possibilities for cost at most, say One \
Dollar.\
\>", "Subsection",
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          {"92.5`", "4.`", "0.`", "0.`", "4.5`", "2.`", "0.`"},
          {"97.33333333333336`", "4.`", "0.`", "0.`", "8.`", 
            "0.6666666666666675`", "0.`"},
          {"98.6035889070147`", "4.`", "0.`", "0.`", "2.232952691680261`", 
            "2.`", "1.3951060358890708`"},
          {"100.`", "1.6470588235294117`", "0.`", "0.`", "6.117647058823529`",
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          {"100.`", "3.7415068699984926`", "0.`", "0.`", 
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          {"100.`", "4.`", "0.`", "0.`", "2.2091586794462197`", "2.`", 
            "1.4798722044728432`"},
          {"100.`", "4.`", "0.`", "0.`", "5.333333333333333`", "2.`", "0.`"},
          {"100.`", "4.`", "0.`", "0.`", "8.`", "0.8000000000000002`", 
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          {"100.`", "4.`", "0.`", "0.49557522123893516`", "8.`", 
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          {"100.`", "4.`", "0.`", "1.8750000000000029`", "2.624999999999996`",
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          {"100.`", "4.`", "0.601503759398496`", "0.`", "3.729323308270678`", 
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          {"100.00000000000001`", "4.`", "0.`", "0.`", "2.179657768651609`", 
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          {"100.00000000000001`", "4.`", "0.`", "0.`", "8.`", 
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          {"100.00000000000001`", "4.`", "0.`", "1.025149700598805`", 
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Cell["\<\
The list is complete in the sense that any feasible menu of cost at \
most One Dollar is a combination of these seventeen (extreme) solutions.  One \
can find menus with Chicken, Eggs or Pork that might be much more desireble \
than the optimal menu.   Also it shows you cannot avoid Oatmeal nor Cherry \
pie within this budget to satisfy the nutritional needs.\
\>", "Text",
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