(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 19132, 550]*) (*NotebookOutlinePosition[ 20178, 586]*) (* CellTagsIndexPosition[ 20090, 580]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Diet Problem", FontColor->RGBColor[0.0557107, 0.137819, 0.517113]], "\nAn Application of Vertex Enumeration\nwith ", StyleBox["cddmathlink2", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " (", StyleBox["MathLink", FontSize->24, FontSlant->"Italic", FontColor->RGBColor[0.0146487, 0.461387, 0.0967727]], " to ", StyleBox["cddlib)", FontColor->RGBColor[0.517113, 0.0273594, 0.0273594]] }], "Title", ImageRegion->{{0, 1}, {0, 1}}, FontSize->27], Cell[TextData[StyleBox["Komei Fukuda, fukuda@ifor.math.ethz.ch\nSwiss Federal \ Institute of Technology, Lausanne and Zurich\nDecember 14, 2002", FontSize->17, FontSlant->"Italic"]], "Subtitle", ImageRegion->{{0, 1}, {0, 1}}], Cell["What's New?", "Section", ImageRegion->{{0, 1}, {0, 1}}, FontSize->20], Cell[TextData[{ StyleBox["cddmathlink2", FontColor->RGBColor[0, 0, 1]], " is a new version of cddmathlink. It comes in two flavors. ", StyleBox["cddmathlink2", FontColor->RGBColor[0, 0, 1]], " is essentially the same as the old cddmathlink which uses floating-point \ arithmetic, while ", StyleBox["cddmathlink2gmp", FontColor->RGBColor[1, 0, 0]], " uses GMP rational exact arithmetic. Thus one can ", StyleBox["compute exactly", FontColor->RGBColor[1, 0, 0]], " with ", StyleBox["cddmathlink2gmp", FontColor->RGBColor[1, 0, 0]], ". I have tested this only with ", StyleBox["Mathematica", FontSlant->"Italic"], " 4.1 on MacOS X. " }], "Text"], Cell[CellGroupData[{ Cell["Connecting cddmathlink2", "Section", InitializationCell->True, ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[TextData[{ "You just put the compiled cddmathlink2 and cddmathlink2gmp on your \ computer in some directory. In this example, the name of the directory is ", StyleBox["\"~/Math\".", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", InitializationCell->True, ImageRegion->{{0, 1}, {0, 1}}], Cell["Off[General::spell1]; Off[General::spell];", "Input", InitializationCell->True, ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell["cddml=Install[\"~/Math/cddmathlink2gmp\"]", "Input", InitializationCell->True, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(LinkObject["/Users/fukuda/Math/cddmathlink2gmp", 3, 3]\)], "Output"], Cell[BoxData[ \( (*If\ you\ prefer\ to\ use\ floating - point\ arithmetic, \(use\ \(cddmathlink2 : cddml\) = Install["\<~/Math/cddmathlink2\>"];\)*) \)], "Input"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["What is Diet Problem?", "Section", ImageRegion->{{0, 1}, {0, 1}}, FontSize->20], 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", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell["\<\ var={\"\",\"Oatmeal\",\"Chicken\",\"Eggs\",\"Milk\",\"Cherry Pie\", \ \t\"Pork Beans\"}; price={\"Price/Ser\", \"3c\", \"24c\", \"13c\", \"9c\", \"20c\", \ \"19c\"}\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}, FontSize->14], Cell[BoxData[ \({"Price/Ser", "3c", "24c", "13c", "9c", "20c", "19c"}\)], "Output"] }, Open ]], Cell["\<\ MatrixForm[dietproblem1= {{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, 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", ImageRegion->{{0, 1}, {0, 1}}, FontSize->14], Cell[CellGroupData[{ Cell["\<\ TableForm[table1=Prepend[Prepend[dietproblem1,var],price]] \ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ TagBox[GridBox[{ {"\<\"Price/Ser\"\>", "\<\"3c\"\>", "\<\"24c\"\>", "\<\"13c\"\>", "\ \<\"9c\"\>", "\<\"20c\"\>", "\<\"19c\"\>"}, {"\<\"\"\>", "\<\"Oatmeal\"\>", "\<\"Chicken\"\>", "\<\"Eggs\"\>", \ "\<\"Milk\"\>", "\<\"Cherry Pie\"\>", "\<\"Pork Beans\"\>"}, {"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", "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"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ m=Transpose[Drop[Transpose[dietproblem1],1]]; b=-First[Transpose[dietproblem1]]; c={3, 24, 13, 9, 20, 19};\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}, FontSize->14], Cell[TextData[{ "By using the build-in LP optimizer of ", StyleBox["Mathematica", FontSlant->"Italic"], ", one can easily compute the optimal solution." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell["lps=LinearProgramming[c, m,b]", "Input", ImageRegion->{{0, 1}, {0, 1}}, FontSize->14], Cell[BoxData[ \({4, 0, 0, 9\/2, 2, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["optvalue= N[c.lps]", "Input", ImageRegion->{{0, 1}, {0, 1}}, FontSize->14], Cell[BoxData[ \(92.5`\)], "Output"] }, Open ]], 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", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell["\<\ TableForm[Join[{var},{Prepend[N[lps],optvalue]}]] \ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}, FontSize->14], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"Oatmeal\"\>", "\<\"Chicken\"\>", "\<\"Eggs\"\>", \ "\<\"Milk\"\>", "\<\"Cherry Pie\"\>", "\<\"Pork Beans\"\>"}, {"92.5`", "4.`", "0.`", "0.`", "4.5`", "2.`", "0.`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Why is the Vertex Enumeration Useful?", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Now we try to do something more reasonable. We use cddmathlink \ fuction AllVertices:\ \>", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell["?AllVertices", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \("AllVertices[m,d+1,A] generates all extreme points (vertices) and \ extreme rays of the convex polyhedron in R^(d+1) given as the solution set to \ an inequality system A x >= 0 where A is an m*(d+1) matrix and \ x=(1,x1,...,xd). The output is {{extlist, linearity}, ecdlist} where extlist \ is the extreme point list and ecdlist is the incidence list. Each vertex \ (ray) has the first component 1 (0). If the convex polyhedron is nonempty \ and has no vertices, extlist is a (nonunique) set of generators of the \ polyhedron where those generators in the linearity list are considered as \ linearity space (of points satisfying A (0, x1, x2, ...., xd) = 0) \ generators."\)], "Print", CellTags->"Info3248898481-6231000"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We can then compute ALL possibilities for cost at most, say One \ Dollar.\ \>", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["BudgetLimit=10111/100;", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(N[BudgetLimit]\)], "Input"], Cell[BoxData[ \(101.11`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ MatrixForm[dietproblem2=Append[dietproblem1, {BudgetLimit, -3, -24, -13, -9, -20, -19}]]\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "0", "0", "0", "0", "0"}, {"0", "0", "1", "0", "0", "0", "0"}, {"0", "0", "0", "1", 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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}, {10111/100, -3, -24, -13, -9, -20, -19}}"\)], "Output"] }, Open ]], Cell["{{extlist,linearity},inclist}=AllVertices[m2,d2,s1];", "Input", ImageRegion->{{0, 1}, {0, 1}}] }, Open ]], Cell[CellGroupData[{ Cell["Length[extlist]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(22\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["vlist=Map[Drop[#,1]&, ToExpression[extlist]]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \({{4, 0, 0, 8, 2\/3, 0}, {4, 0, 0, 9\/2, 2, 0}, {4, 0, 2, 5\/2, 2, 0}, {4, 0, 0, 6844\/3065, 2, 4276\/3065}, {1170957\/331150, 0, 0, 1796723\/827875, 2, 5397633\/3311500}, {4856\/1275, 0, 2, 13421\/5100, 2, 0}, {552\/425, 0, 0, 10807\/1700, 2, 0}, {4, 0, 2, 8531\/2900, 1328\/725, 0}, {4, 488\/9975, 2, 97249\/39900, 2, 0}, {4, 0, 2, 7957\/3500, 2, 122\/875}, {4, 0, 153643\/83500, 458359\/208750, 2, 29818\/104375}, {4, 0, 0, 3903233\/1826250, 1307357\/730500, 6551599\/3652500}, {4, 0, 0, 51416\/23475, 2, 48429\/31300}, {4, 328\/475, 0, 6869\/1900, 2, 0}, {4, 21949\/73875, 0, 848572\/369375, 2, 552209\/492500}, {4, 0, 2, 2311\/900, 2, 0}, {4, 0, 0, 1637\/300, 2, 0}, {10869\/4700, 0, 0, 8, 10421\/9400, 0}, {4, 0, 7931\/11300, 8, 1128\/2825, 0}, {4, 0, 0, 8, 4357\/13900, 7931\/13900}, {4, 7931\/29900, 0, 8, 64249\/119600, 0}, {4, 0, 0, 8, 1711\/2000, 0}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["allsolutions=Union[Map[Prepend[#, N[c.#,3]]&, N[vlist,5]]]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \({{92.5`, 4.`, 0.`, 0.`, 4.5`, 2.`, 0.`}, {97.33333333333333`, 4.`, 0.`, 0.`, 8.`, 0.6666666666666666`, 0.`}, {98.60358890701468`, 4.`, 0.`, 0.`, 2.232952691680261`, 2.`, 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"2.5677777777777777`", "2.`", "0.`"}, {"101.11`", "4.`", "0.`", "2.`", "2.9417241379310344`", "1.8317241379310345`", "0.`"}, {"101.11`", "4.`", "0.04892230576441103`", "2.`", "2.4373182957393484`", "2.`", "0.`"}, {"101.11`", "4.`", "0.26525083612040135`", "0.`", "8.`", "0.5371989966555184`", "0.`"}, {"101.11`", "4.`", "0.29710998307952624`", "0.`", "2.2973184433164127`", "2.`", "1.1212365482233502`"}, {"101.11`", "4.`", "0.6905263157894737`", "0.`", "3.615263157894737`", "2.`", "0.`"}, {"101.11000000000001`", "4.`", "0.`", "2.`", "2.2734285714285716`", "2.`", "0.13942857142857143`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output"] }, Open ]] }, Open ]], Cell["\<\ The list is complete in the sense that any feasible menu of cost at \ most BudgetLimit 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. 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