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McCready\nProfessor and Chair of Chemical Engineering\n\ University of Notre Dame\nNotre Dame IN 46556\nUSA", FontSize->14], "\n\nMark.J.McCready.1@nd.edu\n", ButtonBox["http://www.nd.edu/~mjm/", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/"], None}, ButtonStyle->"Hyperlink"], "\n\n\nIt is copyrighted to the extent allowed by whatever laws pertain to \ the World Wide Web and the Internet.\n\nI would hope that as a professional \ courtesy, this notice remain visible to other users. \nThere is no charge for \ copying and dissemination \n\nVersion: 10/98\nMore recent versions of this \ notebook should be available at the web site:\n", Cell[BoxData[ FormBox[ ButtonBox[\(\(\(http\)\(:\)\) // \(www . nd . edu/\(\(~\)\(mjm\)\)\)/ simplexnotes . nb\), ButtonData:>{ URL[ "http://www.nd.edu/~mjm/simplexnotes.nb"], None}, ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text"], Cell[CellGroupData[{ Cell["Linear programming using the Simplex method", "Subtitle"], Cell[CellGroupData[{ 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Lil001IcW`03001cWg>O0?mcWgicW`005W>O00<007>OLil0og>OOW>O000FLil00`00LimcW`3oLimn Lil001IcW`03001cWg>O0?mcWgicW`005W>O00<007>OLil0og>OOW>O000FLil00`00LimcW`3oLimn Lil00?mcWiMcW`00og>OUg>O003oLinGLil00?mcWiMcW`00og>OUg>O003oLinGLil00?mcWiMcW`00 og>OUg>O003oLinGLil00?mcWiMcW`00og>OUg>O003oLinGLil0011cW`800003Lil00000009cW`@0 0?mcWgacW`004G>O00<007>O000017>O00<007>OLil0og>OO7>O000BLil00`00LimcW`04Lil00`00 LimcW`3oLimkLil0015cW`03001cW`0000IcW`03001cWg>O0?mcWgYcW`0047>O0P0000=cW`000000 0W>O00@007>OLil00?mcWgacW`0067>O0P00og>OOG>O003oLinGLil00?mcWiMcW`00og>OUg>O0000 \ \>"], ImageRangeCache->{{{0, 405.625}, {405.625, 0}} -> {-2.58737, -21.0935, 0.114566, 0.242924}}], Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output", CellLabel->"Out[195]="] }, Open ]], Cell["\<\ We see where the optimimum is!, x1=15, x2=25. Of course we can't \ solve every problem graphically so we need a procedure.\ \>", "Text"], Cell["\<\ Express the problem in an augmented matrix form with the vector \ being {M, x1, x2, x3, x4, x5} and the augmented column {70, 40, 90, 0} the last \ entry (which will be M) being the current value of the objective function. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Start with the real variables set to zero (x1=x2=0) which is a known \ trivial solution. The zero variables will be called nonbasic variables these \ will change as the procedure progresses. To eliminate the need to deal with \ inequalities and because we note that if a constraint is limiting the \ solution, we will be at the limiting value (i.e., it will be an ", StyleBox["equality", FontSlant->"Italic"], ") we will introduce ", StyleBox["slack", FontSlant->"Italic"], " variables to make all of the inequalities into equations. For this \ problem we need three slack variables, x3, x4 and x5. If we start at x1 = x2 \ = 0, then the initial values of the slack variables are x3=70, x4=40 and \ x5=90. This solution is found from inspection. At any step the non zero \ variables will be called basic variables." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Here is the initial augmented matrix. There is some convenience to \ entering it line by line. (We can use a subscripted variable.)\ \>", "Text"], Cell[CellGroupData[{ Cell["z[1][1]= {0,2,1,1,0,0,70}", "Input", CellLabel->"In[227]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`{0, 2, 1, 1, 0, 0, 70}\)], "Output", CellLabel->"Out[227]="], Cell[CellGroupData[{ Cell[BoxData[ \(\(z[1]\)[2] = {0, 1, 1, 0, 1, 0, 40}\)], "Input", CellLabel->"In[228]:="], Cell[BoxData[ \(TraditionalForm\`{0, 1, 1, 0, 1, 0, 40}\)], "Output", CellLabel->"Out[228]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(z[1]\)[3] = {0, 1, 3, 0, 0, 1, 90}\)], "Input", CellLabel->"In[229]:="], Cell[BoxData[ \(TraditionalForm\`{0, 1, 3, 0, 0, 1, 90}\)], "Output", CellLabel->"Out[229]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(z[1]\)[4] = {1, \(-40\), \(-60\), 0, 0, 0, 0}\)], "Input", CellLabel->"In[230]:="], Cell[BoxData[ \(TraditionalForm\`{1, \(-40\), \(-60\), 0, 0, 0, 0}\)], "Output", CellLabel->"Out[230]="] }, Open ]] }, Open ]], Cell["Here we write it as a matrix.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(a1 = Table[\(z[1]\)[i], {i, 1, 4}]\)], "Input", CellLabel->"In[231]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"0", "2", "1", "1", "0", "0", "70"}, {"0", "1", "1", "0", "1", "0", "40"}, {"0", "1", "3", "0", "0", "1", "90"}, {"1", \(-40\), \(-60\), "0", "0", "0", "0"} }, ColumnAlignments->{Decimal}], ")"}], TraditionalForm]], "Output", CellLabel->"Out[231]="] }, Open ]], Cell["\<\ We see, from the objective function, M = 40 x1 + 60 x2, that \ increasing x2 is the way to get the biggest impact. The idea of the simplex \ method is to move only if the objective function will increase and then to \ always take the biggest increase. If instead of looking that the original \ objective function we choose to look at the augmented matrix, we should \ choose to remove the largest negative in the last row and continue doing this \ until there are no more negatives. So, we will move to the next vertex. To do this, which of x3, x4, x5 should \ be set to 0? ANS: The one that represents the strongest constraint on the value of x2! This is found by the getting the smallest (non positive) quotient between the \ x2 column (3rd column) and the b vector (7th column)\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["We make a label for the third column", "Text"], Cell[BoxData[ \(\(kk = 3; \)\)], "Input", CellLabel->"In[232]:="], Cell["\<\ Here we calculate the quotients which are the values of x2 at the \ intersection with the three constraints.\ \>", "Text"], Cell[CellGroupData[{ Cell["ztest=Table[z[1][i][[7]]/z[1][i][[kk]],{i,1,3}]", "Input", CellLabel->"In[233]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`{70, 40, 30}\)], "Output", CellLabel->"Out[233]="], Cell["\<\ We see that we need to use the last constraint. This means that x5 \ will be as small as possible = 0 (i.e., no slack!) and the third constraint \ becomes active. Note that each vertex represents a point where two of the \ variables are 0 and the other three are nonzero. (Why is this?) Thus any \ solution that we examine must have this form. The new value of x2 = 90/3 = 30. \ \>", "Text"], Cell["\<\ We need to identify this active constraint with an index value. \ This command will find the position of the minimum of the smallest value in \ ztest.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(j = \(Position[ztest, \ Min[ztest]]\)[\([1, 1]\)]\)], "Input", CellLabel->"In[234]:="], Cell[BoxData[ \(TraditionalForm\`3\)], "Output", CellLabel->"Out[234]="] }, Open ]] }, Open ]] }, Open ]], Cell["\<\ Now we want a new set of equations that show the change of variables, i.e., showing that x2 has been increased in favor of x5 becoming 0. Keeping x1 = 0 but changing x2 requires that changes be made in x3, x4 and \ x5. x3 = 70 - x2 = 40, x4 = 40 - x2 = 10, x5 = 90 - 3 x2 = 0 So that x1 = 0, x2 = 30, x3 = 40, x4 = 10, x5 =0 is a new basic feasible \ solution. Before we continue on, we need to process the equations above to reflect the \ changes. Since we know that each solution has 2 zero variables and 3 nonzero \ variables, we require that each basic (nonzero) variable has only 1 non zero \ entry in its column in the augmented matrix and this is not in the bottom \ row. This way, each nonzero variable will be the only contribution to the \ value in its equation. The current values of the nonzero variables will be \ immediately evident. The same solution could exist, if say we added two rows \ and messed up our convention, but we would loose the convenience of always \ knowing the solution. Thus we will not do this. To make x2 active (nonzero) with a values at the 3ed constraint, we do a \ Gaussian elimination using the x2 (i.e., kk=3) from the third row, j=3. \ \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Do[If[i != j, \ \(z[2]\)[i] = \(z[1]\)[i] - \(z[1]\)[j]\ \ \(\(z[1]\)[i]\)[\([kk]\)]/\ \(\(z[1]\)[j]\)[\([kk]\)], \(z[2]\)[i] = \(z[1]\)[i]/\(\(z[1]\)[j]\)[\([kk]\)]], {i, 1, 4}]\)], "Input", CellLabel->"In[235]:="], Cell["Here is the new augmented matrix. ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(a2 = Table[\(z[2]\)[i], {i, 1, 4}]\)], "Input", CellLabel->"In[236]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"0", \(5\/3\), "0", "1", "0", \(-\(1\/3\)\), "40"}, {"0", \(2\/3\), "0", "0", "1", \(-\(1\/3\)\), "10"}, {"0", \(1\/3\), "1", "0", "0", \(1\/3\), "30"}, {"1", \(-20\), "0", "0", "0", "20", "1800"} }, ColumnAlignments->{Decimal}], ")"}], TraditionalForm]], "Output", CellLabel->"Out[236]="] }, Open ]], Cell["\<\ We see immediately that x2=30, x3=40 and x4=10 (since x1=x5=0). We \ are thus assured that we at a contrainst intersection as we wish. \ \>", "Text"], Cell[TextData[ "Note that the objective function is 1800 as required and it is expressed by \ the non basic variables (which are 0)"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Now we need to increase x1 as far as possible. Check to see which \ constraint becomes active?\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\(kk = 2; \)\)], "Input", CellLabel->"In[237]:="], Cell[CellGroupData[{ Cell["ztest=Table[z[2][i][[7]]/z[2][i][[kk]],{i,1,3}]", "Input", CellLabel->"In[238]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`{24, 15, 90}\)], "Output", CellLabel->"Out[238]="], Cell["\<\ It will be the second one and we need to identify this active \ constraint with an index value.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(j = \(Position[ztest, \ Min[ztest]]\)[\([1, 1]\)]\)], "Input", CellLabel->"In[239]:="], Cell[BoxData[ \(TraditionalForm\`2\)], "Output", CellLabel->"Out[239]="] }, Open ]] }, Open ]] }, Open ]], Cell[TextData[ "The second constraint will become active first so that x4 will go to 0. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Do[If[i != j, \ \(z[3]\)[i] = \(z[2]\)[i] - \(z[2]\)[j]\ \ \(\(z[2]\)[i]\)[\([kk]\)]/\ \(\(z[2]\)[j]\)[\([kk]\)], \(z[3]\)[i] = \(z[2]\)[i]/\(\(z[2]\)[j]\)[\([kk]\)]], {i, 1, 4}]\)], "Input", CellLabel->"In[240]:="], Cell[CellGroupData[{ Cell[BoxData[ \(a3 = Table[\(z[3]\)[i], {i, 1, 4}]\)], "Input", CellLabel->"In[241]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"0", "0", "0", "1", \(-\(5\/2\)\), \(1\/2\), "15"}, {"0", "1", "0", "0", \(3\/2\), \(-\(1\/2\)\), "15"}, {"0", "0", "1", "0", \(-\(1\/2\)\), \(1\/2\), "25"}, {"1", "0", "0", "0", "30", "10", "2100"} }, ColumnAlignments->{Decimal}], ")"}], TraditionalForm]], "Output", CellLabel->"Out[241]="] }, Open ]], Cell["\<\ So the objective function is 2100! There are no remaining \ \"negatives\" in the last row so that no improvement can be attained. So we are done! For x4=x5=0, the solution for {x1,x2,x3} = {15, 25, 15}. We \ see that x2 was reduced a little to get a big gain from x1. Now x4 and x5=0 which \ correspond to the constrints, x1 + x2 <=40 x1 + 3 x2 <=90 being active. Here is a check of the objective function!\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["15*40+25*60", "Input", CellLabel->"In[211]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`2100\)], "Output", CellLabel->"Out[211]="] }, Open ]], Cell[TextData[{ "Of course, ", StyleBox["Mathematica", FontSlant->"Italic"], " did not really need our help" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ConstrainedMax[ 40\ x1\ + \ 60\ x2, {2\ x1\ + \ x2\ <= 70, \n\t\t x1\ + \ x2\ <= 40, \ x1\ + \ 3\ x2 <= 90}, {x1, x2}]\)], "Input", CellLabel->"In[212]:="], Cell[BoxData[ \(TraditionalForm\`{2100, {x1 \[Rule] 15, x2 \[Rule] 25}}\)], "Output", CellLabel->"Out[212]="] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for Macintosh", ScreenRectangle->{{0, 1152}, {0, 850}}, WindowToolbars->"RulerBar", CellGrouping->Manual, WindowSize->{547, 656}, WindowMargins->{{123, Automatic}, {Automatic, 68}}, PrintingCopies->1, PrintingPageRange->{1, 11}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, StyleDefinitions -> "mjmClassic.nb", MacintoshSystemPageSetup->"\<\ 00<0004/0B`000002mT8o?mooh<" ] (*********************************************************************** Cached data follows. 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