(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 8.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 835, 17] NotebookDataLength[ 898657, 17650] NotebookOptionsPosition[ 877721, 17005] NotebookOutlinePosition[ 881035, 17113] CellTagsIndexPosition[ 880887, 17105] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Optimizaci\[OAcute]n lineal y no lineal con ", StyleBox["Mathematica", FontSlant->"Italic"], " " }], "Title", CellChangeTimes->{{3.4650439076578093`*^9, 3.465043911136609*^9}, { 3.4652829785934*^9, 3.465283009731*^9}, {3.4652830668738003`*^9, 3.4652830677942*^9}, {3.4943347875807652`*^9, 3.494334789924899*^9}, { 3.497937007406066*^9, 3.4979370103960695`*^9}, {3.5258684756706705`*^9, 3.5258684791650763`*^9}, {3.5263703344999113`*^9, 3.5263703350147123`*^9}, 3.533014942025239*^9}], Cell["\<\ M\[EAcute]todos de Optimizaci\[OAcute]n ADE. Pr\[AAcute]cticas. Dpto de \ Econom\[IAcute]a e Historia Econ\[OAcute]mica. Universidad de Salamanca. \ \>", "Department", CellChangeTimes->{{3.494334840667802*^9, 3.4943348428299255`*^9}, { 3.5258703775254116`*^9, 3.525870425074295*^9}, 3.562827682970863*^9}], Cell["\<\ Guillermo S\[AAcute]nchez (guillermo@usal.es). \ \>", "Department", CellChangeTimes->{{3.4837890108305173`*^9, 3.4837890268484335`*^9}, { 3.4841327900220284`*^9, 3.4841328250710335`*^9}, {3.4938803434212866`*^9, 3.493880345854891*^9}, {3.493880437033063*^9, 3.4938805109303927`*^9}, { 3.4943348349314737`*^9, 3.494334836235548*^9}, 3.4943348689874215`*^9, { 3.5258703520807667`*^9, 3.5258703665585923`*^9}, {3.533014944958044*^9, 3.533014947812849*^9}}], Cell["\<\ \[CapitalUAcute]ltima actualizaci\[OAcute]n: 2012-11-25\ \>", "Date", CellChangeTimes->{{3.4804950674592*^9, 3.4804950764142*^9}, { 3.4840205136163263`*^9, 3.484020514579381*^9}, 3.4841328389768286`*^9, { 3.484197001843934*^9, 3.4841970035970345`*^9}, {3.484484113600335*^9, 3.484484115375437*^9}, {3.4845658596179986`*^9, 3.4845658603870425`*^9}, { 3.4847264654135156`*^9, 3.4847264659175444`*^9}, {3.4850951298189125`*^9, 3.485095130463949*^9}, {3.4852652346880302`*^9, 3.485265236231119*^9}, { 3.4938784575211115`*^9, 3.49387846220112*^9}, {3.494334845419073*^9, 3.494334847339183*^9}, {3.497934073980075*^9, 3.4979340845396786`*^9}, { 3.525868431990594*^9, 3.5258684404146085`*^9}, {3.5263732587406473`*^9, 3.52637326023825*^9}, {3.5330128140122657`*^9, 3.5330128204756355`*^9}, { 3.5330544399715567`*^9, 3.533054442717161*^9}, {3.5628274011227555`*^9, 3.5628274059276223`*^9}}], Cell[CellGroupData[{ Cell["\<\ Funciones de optimizaci\[OAcute]n disponibles en Mathematica\ \>", "Section", CellChangeTimes->{ 3.525255611161892*^9, 3.5252556466479216`*^9, {3.526370050283012*^9, 3.526370051811815*^9}}], Cell[TextData[{ "Los problemas de optimizaci\[OAcute]n matematicamente consisten en lo \ siguiente: Para una funci\[OAcute]n de ", Cell[BoxData[ FormBox["n", TraditionalForm]], "InlineFormula"], " variables, el \[AAcute]rea que cumple con las restricciones del problema \ en una regi\[OAcute]n de un espacio ", Cell[BoxData[ FormBox["n", TraditionalForm]], "InlineFormula"], "\[Hyphen]dimensional donde cada desigualdad , o restricci\[OAcute]n, es un \ plano, en el espacio ", Cell[BoxData[ FormBox["n", TraditionalForm]], "InlineFormula"], "\[Hyphen]dimensional, que l\[IAcute]mita uno de los lados de la regi\ \[OAcute]n. " }], "Text", CellChangeTimes->{{3.493878620042197*^9, 3.493878658745865*^9}, { 3.5330151426565924`*^9, 3.5330151549806137`*^9}}], Cell[TextData[{ "Para resolver este tipo de problemas con ", StyleBox["Mathematica", FontSlant->"Italic"], " disponemos, entre otras, de las funciones ", StyleBox["NMaximize/Maximize", "MR"], " y ", StyleBox["NMinimize/Minimize", "MR"], "." }], "Text", CellChangeTimes->{{3.493878620042197*^9, 3.493878658745865*^9}, { 3.5330151426565924`*^9, 3.5330151549806137`*^9}}], Cell[TextData[{ StyleBox["NMaximize[{", "MR"], StyleBox["f", "TI"], StyleBox[",", "MR"], " ", StyleBox["restricciones}", FontSlant->"Italic"], ", {", StyleBox["variables", FontSlant->"Italic"], "}", StyleBox["]", "MR"], " encuentra, por m\[EAcute]todos n\[UAcute]mericos, el m\[AAcute]ximo de f \ que cumplen las condiciones especificadas en las restricciones para las \ variables definidas. Tambien se puede utilizar para ", StyleBox["NMaximize[", "MR"], StyleBox["f", "TI"], ", {", StyleBox["variables", FontSlant->"Italic"], "}] para calcularse m\[AAcute]ximos globales sin restricciones-" }], "Text", CellChangeTimes->{{3.4649735289418*^9, 3.4649735432314*^9}, { 3.4649735996098003`*^9, 3.4649736302482*^9}, {3.4649737139119997`*^9, 3.464973733178*^9}, {3.4649738423*^9, 3.4649738944196*^9}, { 3.4649739250892*^9, 3.4649740392032003`*^9}, {3.4652129667792153`*^9, 3.4652130025032153`*^9}, {3.493878717557968*^9, 3.493878853387407*^9}, { 3.525869784580969*^9, 3.5258697989329944`*^9}}, Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[{ "Para calcular m\[IAcute]nimos se empleara ", StyleBox["NMinimize[{", "MR"], StyleBox["f", "TI"], StyleBox[",", "MR"], " ", StyleBox["restricciones}", FontSlant->"Italic"], ", {", StyleBox["variables", FontSlant->"Italic"], "}", StyleBox["]. Maximize/Minimize", "MR"], " aplican metodos analiticos y buscan m\[AAcute]ximos o m\[IAcute]nimos \ globales exactos, cuando es posible. " }], "Text", CellChangeTimes->{{3.4649735289418*^9, 3.4649735432314*^9}, { 3.4649735996098003`*^9, 3.4649736302482*^9}, {3.4649737139119997`*^9, 3.464973733178*^9}, {3.4649738423*^9, 3.4649738944196*^9}, { 3.4649739250892*^9, 3.4649740392032003`*^9}, {3.4652129667792153`*^9, 3.4652130025032153`*^9}, {3.493878717557968*^9, 3.493878853387407*^9}, { 3.525869784580969*^9, 3.5258698871199493`*^9}}, Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[{ "Para la funci\[OAcute]n ", Cell[BoxData[ FormBox[ RowBox[{"x", "+", "y"}], TraditionalForm]], "InlineFormula"], " con la restricciones ", Cell[BoxData[ FormBox[ RowBox[{"x", "<", "1"}], TraditionalForm]], "InlineFormula"], " y ", Cell[BoxData[ FormBox[ RowBox[{"y", "<", "2"}], TraditionalForm]], "InlineFormula"], " , definido no negativo {0\[LessSlantEqual]x, 0\[LessSlantEqual]y}, podemos \ calcular el m\[AAcute]ximo como sigue " }], "Text", CellChangeTimes->{{3.4650406051066093`*^9, 3.4650406490674095`*^9}, { 3.465041100562609*^9, 3.465041141247409*^9}, {3.465041294720209*^9, 3.465041315452609*^9}, {3.465041411720209*^9, 3.465041415854209*^9}, { 3.5258678859116344`*^9, 3.5258678876900377`*^9}}, CellTags->{"S3.9.9", "9.5"}], Cell[CellGroupData[{ Cell["\<\ NMaximize[{x + y, 0\[LessSlantEqual]x\[LessSlantEqual]1, 0\[LessSlantEqual]y\ \[LessSlantEqual]2}, {x, y}]\ \>", "Input", CellChangeTimes->{{3.4649740667060003`*^9, 3.4649740923992*^9}, { 3.465041155318609*^9, 3.4650412265170093`*^9}, {3.465042037810609*^9, 3.465042055859809*^9}, {3.4938789977498646`*^9, 3.4938790009166703`*^9}}, CellTags->"S3.9.9"], Cell[BoxData[ RowBox[{"{", RowBox[{"3.`", ",", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "1.`"}], ",", RowBox[{"y", "\[Rule]", "2.`"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.5258688306180935`*^9, 3.533014964817879*^9}, CellTags->"S3.9.9"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Maximize[{x + y, 0\[LessSlantEqual]x\[LessSlantEqual]1, 0\[LessSlantEqual]y\ \[LessSlantEqual]2}, {x, y}]\ \>", "Input", CellChangeTimes->{{3.4649740667060003`*^9, 3.4649740923992*^9}, { 3.465041155318609*^9, 3.4650412265170093`*^9}, {3.465042037810609*^9, 3.465042055859809*^9}, 3.4938788746970444`*^9, {3.493878984755042*^9, 3.4938789878750477`*^9}}, CellTags->"S3.9.9"], Cell[BoxData[ RowBox[{"{", RowBox[{"3", ",", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "1"}], ",", RowBox[{"y", "\[Rule]", "2"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.525868830664894*^9, 3.5330149650830793`*^9}, CellTags->"S3.9.9"] }, Open ]], Cell["\<\ Vemos que el maximo est\[AAcute] en {x \[Rule] 1, y \[Rule] 2} y es 3. \ Podemos comprobarlo (recordemos que /. se utiliza para reemplazamientos}\ \>", "Text", CellChangeTimes->{{3.4650414489886093`*^9, 3.465041458208209*^9}, { 3.465041545162609*^9, 3.4650415699510093`*^9}, {3.4650416072506094`*^9, 3.4650416845330095`*^9}, {3.465041727557809*^9, 3.465041743407409*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"x", " ", "+", " ", "y"}], "/.", RowBox[{"{", 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Editable->False, SelectWithContents->True]], "Output", CellChangeTimes->{3.525868830867694*^9, 3.53301496536388*^9}] }, Open ]], Cell["Podemos copiar la entrada y modificarla si es necesario.", "Text", CellChangeTimes->{{3.5258681795665503`*^9, 3.525868200532987*^9}, { 3.525868233355445*^9, 3.525868257519887*^9}}], Cell["\<\ El m\[IAcute]nimo para las mismas condiciones podemos calcularlo como sigue :\ \>", "Text", CellChangeTimes->{{3.4650419447254095`*^9, 3.465041965067809*^9}, { 3.525867688274887*^9, 3.525867689148489*^9}}], Cell[CellGroupData[{ Cell["\<\ NMinimize[{x + y, 0\[LessSlantEqual]x<1, 0\[LessSlantEqual]y<2}, {x, y}]\ \>", "Input", CellChangeTimes->{{3.4649740667060003`*^9, 3.4649740923992*^9}, { 3.465041155318609*^9, 3.4650412265170093`*^9}, {3.465042037810609*^9, 3.465042079603009*^9}}, CellTags->"S3.9.9"], Cell[BoxData[ RowBox[{"{", RowBox[{"0.`", ",", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "0.`"}], ",", RowBox[{"y", "\[Rule]", "0.`"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.5258688309144945`*^9, 3.5330149694842873`*^9}, CellTags->"S3.9.9"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Minimize[{x + y, 0\[LessSlantEqual]x<1, 0\[LessSlantEqual]y<2}, {x, y}]\ \>", "Input", CellChangeTimes->{{3.4649740667060003`*^9, 3.4649740923992*^9}, { 3.465041155318609*^9, 3.4650412265170093`*^9}, {3.465042037810609*^9, 3.465042079603009*^9}, 3.4938789122151103`*^9}, CellTags->"S3.9.9"], Cell[BoxData[ RowBox[{"{", RowBox[{"0", ",", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "0"}], ",", RowBox[{"y", "\[Rule]", "0"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.5258688309456944`*^9, 3.533014969718288*^9}, CellTags->"S3.9.9"] }, Open ]], Cell[TextData[{ "Veamos el problema graficamente: representamos el plano", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"x", " ", "+", " ", "y"}]}], TraditionalForm]]], " y lo acotamos en la region: {", Cell[BoxData[ FormBox[ RowBox[{"0", "\[LessSlantEqual]", "x", "\[Precedes]", "1"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{"0", "\[LessSlantEqual]", "y", "\[Precedes]", "2"}], TraditionalForm]]], "}. Vemos que en el grafico el maximo esta en (1,2) y (0,0). Pruebe a pulsar \ con el rat\[OAcute]n sobre el centro del gr\[AAcute]fico y desplazelo para \ verlo sobre distinta perspectivas. El simbolo && correspode a la proposici\ \[OAcute]n l\[OAcute]gica Y (o AND). En este caso indica que deben cumplirse \ simultaneamente las dos restricciones indicadas. Pueden incluirse tantas \ restricciones como se desee." }], "Text", CellChangeTimes->{{3.4650417732814093`*^9, 3.465041914289809*^9}, { 3.465042094657009*^9, 3.4650421301158094`*^9}, {3.4650425884438095`*^9, 3.465042657645409*^9}, {3.4652767993122*^9, 3.4652768409642*^9}, { 3.4652768742702*^9, 3.4652770378986*^9}, {3.4652850593993998`*^9, 3.4652850712553997`*^9}, 3.4653038799642*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"x", " ", "+", " ", "y"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "2"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"y", ",", " ", RowBox[{"-", "1"}], ",", " ", "2"}], "}"}], ",", " ", RowBox[{"RegionFunction", " ", "->", " ", RowBox[{"Function", "[", RowBox[{ RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}], ",", RowBox[{ RowBox[{"0", "\[LessEqual]", "x", "<", "1"}], "&&", RowBox[{"0", "\[LessEqual]", "y", "<", "2"}]}]}], "]"}]}], ",", RowBox[{"AxesLabel", "\[Rule]", "Automatic"}], ",", RowBox[{"Mesh", "\[Rule]", 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FormBox[ RowBox[{" ", RowBox[{ RowBox[{"x", " ", "+", " ", "y"}], " ", "\[GreaterEqual]", " ", "26"}]}], TraditionalForm], "\n", FormBox[ RowBox[{" ", RowBox[{ RowBox[{"x", " ", "\[GreaterEqual]", " ", "3"}], ",", " ", RowBox[{"y", " ", "\[GreaterEqual]", " ", "4"}]}]}], TraditionalForm]}], "InlineMath"]], "Text", CellChangeTimes->{{3.4652822032412*^9, 3.4652823609572*^9}, { 3.5253330154685965`*^9, 3.5253330209369097`*^9}}], Cell[TextData[{ Cell[BoxData[ StyleBox[ ButtonBox["NMinimize", BaseStyle->"Link", ButtonData->"paclet:ref/NMinimize"], FontFamily->"Courier"]], "InlineFormula"], " vimos que da una aproximaci\[OAcute]n num\[EAcute]rica (suficientemente \ buena para la mayoria de los problemas pr\[AAcute]cticos)" }], "Text", CellChangeTimes->{{3.4652823886628*^9, 3.4652824415624*^9}, { 3.465282587516*^9, 3.4652826130220003`*^9}, {3.4661544395109997`*^9, 3.466154440329*^9}, {3.4943317561436577`*^9, 3.4943317756124926`*^9}}, CellID->141146064], Cell[CellGroupData[{ Cell[BoxData[ 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Recordemos que prueba a utilizar m\[EAcute]todos anal\[IAcute]ticos para \ encontrar la soluci\[OAcute]n exacta." }], "Text", CellChangeTimes->{{3.4652822032412*^9, 3.46528238495*^9}, {3.465282617078*^9, 3.465282702566*^9}, {3.4943317848477087`*^9, 3.494331866623052*^9}, { 3.5266421164578285`*^9, 3.526642117313878*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Minimize", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", "+", RowBox[{"2", " ", "y"}]}], ",", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "5"}], "x"}], "+", "y"}], "\[Equal]", "7"}], " ", "&&", RowBox[{ RowBox[{"x", "+", "y"}], "\[GreaterEqual]", "26"}], "&&", " ", RowBox[{"x", "\[GreaterEqual]", "3"}], "&&", RowBox[{"y", "\[GreaterEqual]", "4"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}]}], "]"}]], "Input", CellID->138525404], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox["293", "6"], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", FractionBox["19", "6"]}], ",", RowBox[{"y", "\[Rule]", FractionBox["137", "6"]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.5330139030638895`*^9, 3.5330139474393682`*^9, 3.53301497093509*^9}, ImageSize->{171, 32}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Open ]], Cell[TextData[{ "Tambi\[EAcute]n podemos usar la ", Cell[BoxData[ ButtonBox["FindMinimum", BaseStyle->"Link", ButtonData->"paclet:ref/FindMinimum"]]] }], "Text", CellChangeTimes->{{3.465360567342798*^9, 3.465360619025598*^9}, { 3.5263734258325405`*^9, 3.5263734270805435`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindMinimum", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", "+", RowBox[{"2", " ", "y"}]}], ",", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "5"}], "x"}], "+", "y"}], "\[Equal]", "7"}], " ", "&&", RowBox[{ RowBox[{"x", "+", "y"}], "\[GreaterEqual]", "26"}], "&&", " ", RowBox[{"x", "\[GreaterEqual]", "3"}], "&&", RowBox[{"y", "\[GreaterEqual]", "4"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.465360487517598*^9, 3.465360493383198*^9}, { 3.465360543646398*^9, 3.465360557171598*^9}, 3.526373594858838*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"48.83333333333333`", ",", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "3.1666666666666665`"}], ",", RowBox[{"y", "\[Rule]", "22.833333333333332`"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.465360559542798*^9, 3.53301390320429*^9, 3.533013947486168*^9, 3.5330149710286903`*^9}] }, Open ]], Cell[TextData[StyleBox["\[DownQuestion]C\[OAcute]mo forzar a que una o todas \ las variables sean enteras?", FontWeight->"Bold"]], "Text", CellChangeTimes->{{3.5263724348876004`*^9, 3.526372465713255*^9}, { 3.526372589920673*^9, 3.526372591433875*^9}}, CellID->1896901598], Cell["\<\ En algunos problemas de optimizaci\[OAcute]n se requiere que las soluciones \ sean enteras, puede hacerse como sigue (en este caso solo se requiere que \ sea entera la variable y):\ \>", "Text", CellChangeTimes->{{3.5263724761496725`*^9, 3.5263725274113626`*^9}, { 3.5263733162579484`*^9, 3.5263733173187504`*^9}, {3.5263736390849156`*^9, 3.526373657960949*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindMinimum", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", "+", RowBox[{"2", " ", "y"}]}], ",", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "5"}], "x"}], "+", "y"}], "\[Equal]", "7"}], " ", "&&", RowBox[{ RowBox[{"x", "+", "y"}], "\[GreaterEqual]", "26"}], "&&", " ", RowBox[{"x", "\[GreaterEqual]", "3"}], "&&", RowBox[{"y", "\[GreaterEqual]", "4"}], "&&", RowBox[{"y", "\[Element]", "Integers"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.465360487517598*^9, 3.465360493383198*^9}, { 3.465360543646398*^9, 3.465360557171598*^9}, {3.526373594858838*^9, 3.526373623531688*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"49.2`", ",", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "3.2`"}], ",", RowBox[{"y", "\[Rule]", "23"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.5263736303021*^9, 3.53301390340709*^9, 3.5330139476109686`*^9, 3.5330149713406906`*^9}] }, Open ]], Cell[TextData[{ "El mismo problema utilizando ", Cell[BoxData[ StyleBox[ ButtonBox["LinearProgramming", BaseStyle->"Link", ButtonData->"paclet:ref/LinearProgramming"], FontFamily->"Courier"]], "InlineFormula"], ". Esta funci\[OAcute]n es muy \[UAcute]til cuando se trabaja con un gran n\ \[UAcute]mero de varibles pues simplifica enormente la entrada de datos." }], "Text", CellChangeTimes->{{3.4650435454882092`*^9, 3.4650436182154093`*^9}, { 3.4652770907358*^9, 3.4652771376918*^9}, {3.465282816914*^9, 3.4652828583476*^9}, {3.466154276939*^9, 3.4661543122390003`*^9}, { 3.4669487942153664`*^9, 3.4669487946209664`*^9}}, CellTags->{"S3.9.9", "9.19"}], Cell[TextData[{ "La sintaxis es la siguente: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"LinearPrograming", "[", RowBox[{"c", ",", " ", "m", ",", " ", "b"}], "]"}], " "}], TraditionalForm]], FontSlant->"Italic"], "que encuentra el vector ", StyleBox["x", FontWeight->"Bold"], " que minimiza ", StyleBox["c x", FontWeight->"Bold"], " sujeto a las restricciones ", StyleBox["m.x", FontWeight->"Bold"], " \[GreaterEqual] ", StyleBox["b", FontWeight->"Bold"], " y ", StyleBox["x ", FontWeight->"Bold"], "\[GreaterEqual] 0" }], "Text", 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negativas" }], "Subsubsection", CellChangeTimes->{{3.4938793465976777`*^9, 3.4938793729773235`*^9}, 3.5258699971001425`*^9}], Cell[BoxData[ RowBox[{ RowBox[{"Clear", "[", "\"\\"", "]"}], " "}]], "Input", CellChangeTimes->{{3.5044182991283803`*^9, 3.504418305961192*^9}, { 3.5044441662228003`*^9, 3.5044441677827997`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"var", "=", RowBox[{"{", RowBox[{ SubscriptBox["x", "1"], ",", SubscriptBox["x", "2"], ",", SubscriptBox["x", "3"]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.4652779235418*^9, 3.4652779244778*^9}}], Cell["\<\ Planteamos la funci\[OAcute]n a optimizar o funci\[OAcute]n objetivo (en \ nuestro caso se trata de maximizar el beneficio)\ \>", "Text", CellChangeTimes->{{3.5266421824006004`*^9, 3.5266421861768165`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"fo", " ", "=", " ", RowBox[{ RowBox[{"51", SubscriptBox["x", "1"]}], "+", RowBox[{"30", SubscriptBox["x", "2"]}], "+", RowBox[{"25", " ", SubscriptBox["x", "3"]}]}]}], ";"}]], "Input", 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Eso puede \ hacerse escribiendolas de una en una, como en el ejemplo anterior, o de forma \ mucho m\[AAcute]s sencilla como se muestra a continuaci\[OAcute]n (utilizamos \ el \[UAcute]ltimo ejemplo)\ \>", "Text", CellChangeTimes->{{3.526372647001173*^9, 3.526372746950548*^9}, { 3.526372822907082*^9, 3.526372859317546*^9}, {3.5263732866802964`*^9, 3.5263733073035326`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"rn", "=", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"#", "\[GreaterEqual]", " ", "0"}], ")"}], "&"}], ",", " ", "var"}], "]"}]}]], "Input", CellChangeTimes->{{3.526372757979768*^9, 3.526372761973375*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["x", "1"], "\[GreaterEqual]", "0"}], ",", RowBox[{ SubscriptBox["x", "2"], "\[GreaterEqual]", "0"}], ",", RowBox[{ SubscriptBox["x", "3"], "\[GreaterEqual]", "0"}]}], "}"}]], "Output", CellChangeTimes->{3.526372773938596*^9, 3.5330139041090913`*^9, 3.533013949624372*^9, 3.5330149716214914`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NMaximize", "[", RowBox[{ RowBox[{"{", RowBox[{"fo", ",", "r1", ",", " ", "r2", ",", " ", "rn"}], "}"}], ",", "var"}], "]"}]], "Input", CellChangeTimes->{{3.5263727884310217`*^9, 3.526372790303025*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"3600.0000000000005`", ",", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["x", "1"], "\[Rule]", "0.`"}], ",", RowBox[{ SubscriptBox["x", "2"], "\[Rule]", "70.00000000000001`"}], ",", RowBox[{ SubscriptBox["x", "3"], "\[Rule]", "60.`"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.526372791020626*^9, 3.5330139042494917`*^9, 3.533013949795972*^9, 3.5330149716838913`*^9}] }, Open ]], Cell[BoxData[ InterpretationBox[Cell["\t", "ExampleDelimiter"], $Line = 0; 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Si se deja la \ opci\[OAcute]n por defecto los colores m\[AAcute]s oscuros representan los \ valores m\[AAcute]s peque\[NTilde]os y los m\[AAcute]s claros lo m\[AAcute]s \ elevados. Para entenderlo mejor comparelo con el gr\[AAcute]fico anterior. Si \ se desplaza por el gr\[AAcute]fico el cursor del rat\[OAcute]n por los \ limites entre contornos se ver\[AAcute] los valores que va tomando la funci\ \[OAcute]n. Los puntos unidos por la misma l\[IAcute]nea corresponden a \ aquellos en los que la funci\[OAcute]n toma el mismo valor.\ \>", "Text", CellChangeTimes->{{3.4652757829088*^9, 3.46527581063*^9}, { 3.4652760626792*^9, 3.465276190646*^9}, {3.4652763470994*^9, 3.4652764061766*^9}, {3.4652766397398*^9, 3.4652766563382*^9}, { 3.466154235152*^9, 3.4661542359969997`*^9}, {3.4669489116209664`*^9, 3.466948923055766*^9}, {3.5252592703467274`*^9, 3.525259280429304*^9}, 3.525259317178406*^9, {3.5263737285198727`*^9, 3.526373734915884*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Ejercicios sobre aplicaciones lineales", "Section", CellChangeTimes->{{3.49433349702695*^9, 3.4943335081225843`*^9}, { 3.5263710678727994`*^9, 3.526371101210058*^9}}], Cell["Se sugiere resolver estos mismos ejemplos con Solver", "Item", CellChangeTimes->{{3.5628274586405807`*^9, 3.5628274853948193`*^9}}], Cell[TextData[{ "Atajos de teclado: Los sub\[IAcute]ndices los puede escribir utilizando ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"[", "Crt", "]"}], "+", RowBox[{"[", "-", "]"}]}], TraditionalForm]]], ". Ejemplo: para escribir ", Cell[BoxData[ FormBox[ SubscriptBox["x", "12"], TraditionalForm]]], " pulse: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"[", "Crt", "]"}], "+", RowBox[{"[", "-", "]"}]}], TraditionalForm]]], "12 a continuacion pulse la tecla [\[Rule]] o utilice el cursor para salir \ del sub\[IAcute]ndices. Para escribir exponentes puede usar ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"[", "Crt", "]"}], "+", RowBox[{"[", "6", "]"}]}], TraditionalForm]]], " , por ejemplo, para escribir ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "a"], TraditionalForm]]], " pulse: x", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"[", "Crt", "]"}], "+", RowBox[{"[", "6", "]"}]}], TraditionalForm]]], "a." }], "Text", CellChangeTimes->{{3.497937193684371*^9, 3.4979374992968245`*^9}, { 3.497937532290904*^9, 3.4979377288371954`*^9}, 3.5263710487939663`*^9, { 3.526642358915635*^9, 3.5266423746092625`*^9}}, Background->RGBColor[0.9, 1, 1]], Cell[CellGroupData[{ Cell["\<\ Ejercicio 1.- Fabricaci\[OAcute]n de juguetes \ \>", "Subsection", CellChangeTimes->{ 3.4943347614092684`*^9, {3.4943350930912395`*^9, 3.4943350935952682`*^9}, 3.526371175232188*^9}], Cell["\<\ Una factor\[IAcute]a fabrica dos tipos de juguetes de madera:soldados y \ trenes. El beneficio que obtiene es de 3 euros por cada soldado y de 2 euros \ por cada tren.Un soldado necesita 2 horas de acabado y 1 hora de carpinter\ \[IAcute]a,mientras que cada tren necesita 1 hora de acabado y 1 hora de \ carpinter\[IAcute]a.Cada semana se dispone de 100 horas de acabado y 80 de \ carpinter\[IAcute]a. Por demanda del mercado no se venden m\[AAcute]s de 40 \ soldados cada semana. a) \[DownQuestion]Cuantas unidades debemos de fabricar de cada tipo de \ juguete para maximizar el beneficio?\ \>", "Text", CellChangeTimes->{3.4652784922398*^9}], Cell["\<\ b) \[DownQuestion] Cu\[AAcute]nto estar\[IAcute]amos dispuestos a pagar por \ 10 horas m\[AAcute]s de carpinter\[IAcute]a?\ \>", "Text"], Cell[CellGroupData[{ Cell["Sol", "Subsubsection", CellChangeTimes->{{3.49433511722762*^9, 3.494335117675646*^9}}], Cell["Sol apdo. a)", "Text"], Cell["\<\ Variables: X1, X2 : N\[UAcute]mero de soldados y trenes a fabricar\ \>", "Text", CellChangeTimes->{3.465278486889*^9, 3.5330130145269175`*^9}], Cell[BoxData[ RowBox[{ RowBox[{"var", "=", " ", RowBox[{"{", RowBox[{"x1", ",", "x2"}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.5330129385497837`*^9, 3.533012969484638*^9}}], Cell["Funcion objetivo (a maximizar): 3 X1 + 2 X2", "Text", CellChangeTimes->{ 3.465278486889*^9, {3.5330130145269175`*^9, 3.5330130210331287`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"fo", "=", " ", RowBox[{ RowBox[{"3", " ", "x1"}], "+", RowBox[{"2", " ", "x2"}]}]}], ";"}]], "Input", CellChangeTimes->{{3.5330129385497837`*^9, 3.5330130049017005`*^9}}], Cell["\<\ Restricciones:2 X1 + X2 \[LessEqual]100; X1 + X2\[LessEqual] 80; X1 \ \[LessEqual] 40; X1 ,X2\[GreaterEqual]0\ \>", "Text", CellChangeTimes->{ 3.465278486889*^9, {3.5330130145269175`*^9, 3.5330130210331287`*^9}, 3.533013174088998*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"r1", "=", " ", RowBox[{ RowBox[{ RowBox[{"2", " ", "x1"}], "+", "x2"}], "\[LessEqual]", "100"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"r2", "=", " ", RowBox[{ RowBox[{"x1", "+", "x2"}], "\[LessEqual]", "80"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"r3", "=", RowBox[{"x1", "\[LessEqual]", "40"}]}], ";"}]}], "Input", CellChangeTimes->{{3.533013042732767*^9, 3.5330130976760635`*^9}, 3.533013169986191*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NMaximize", "[", RowBox[{ RowBox[{"{", RowBox[{"fo", ",", "r1", ",", "r2", ",", "r3", ",", RowBox[{"x1", "\[GreaterEqual]", "0"}], ",", RowBox[{"x2", "\[GreaterEqual]", "0"}]}], "}"}], ",", " ", "var"}], "]"}]], "Input", CellChangeTimes->{{3.5330131075050807`*^9, 3.533013180469409*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"180.`", ",", RowBox[{"{", RowBox[{ RowBox[{"x1", "\[Rule]", "20.`"}], ",", RowBox[{"x2", "\[Rule]", "60.`"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.5330131881758227`*^9, 3.5330136232351933`*^9, 3.5330139097729015`*^9, 3.5330139550943823`*^9, 3.5330149765979*^9}] }, Open ]], Cell["\<\ Sol apdo. b). Es necesario modificar una de las restricciones -\[DownQuestion]cual?-y ver \ que beneficio se obtiene. Se compara con el obtenido en el apdo a) y de ah\ \[IAcute] puede deducirse directamente al soluci\[OAcute]n.\ \>", "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Ejercicio 2 .- Lineas de fabricaci\[OAcute]n \ \>", "Subsection", CellChangeTimes->{{3.4652966908354354`*^9, 3.4652967098674355`*^9}, 3.494334765337493*^9, {3.4943351894107485`*^9, 3.4943351932909703`*^9}}], Cell["\<\ Una empresa produce dos productos en dos l\[IAcute]neas de \ fabricaci\[OAcute]n diferentes.La primera l\[IAcute]nea tiene una capacidad \ de producci\[OAcute]n de 60 unidades y la segunda de 50. Producir el primer \ producto requiere 1 hora mientras que producir el segundo requiere 2 h.El \ total de horas disponibles es de 120. El beneficio que se obtiene de los dos \ productos es de 20 \[Euro] y 30 \[Euro] respectivamente.\ \>", "Text", CellChangeTimes->{ 3.4652967790690355`*^9, {3.4652971714090357`*^9, 3.4652971720018353`*^9}}], Cell["\<\ a) \[DownQuestion]Cuantas unidades hay que fabricar de cada producto para que \ el beneficio sea m\[AAcute]ximo?. \[DownQuestion]Cual ser\[AAcute] el \ beneficio?\ \>", "Text", CellChangeTimes->{{3.4652967790690355`*^9, 3.4652967846538353`*^9}}], Cell["\<\ b) En la soluci\[OAcute]n encontrada: \[DownQuestion]Se consumen todas las \ horas de trabajo disponibles? \[DownQuestion]Se agota la capacidad de \ producci\[OAcute]n de las dos l\[IAcute]neas de producci\[OAcute]n?\ \>", "Text", CellChangeTimes->{{3.4652967790690355`*^9, 3.4652967846538353`*^9}}], Cell[CellGroupData[{ Cell["Sol", "Subsubsection", CellChangeTimes->{{3.4979339359201784`*^9, 3.4979339366852217`*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.466151159341*^9, 3.46615116226*^9}, 3.466151198587*^9}], Cell["\<\ a) MAX 20 X1 + 30 X2 Restricciones: X1 <= 60; X2 <= 50; X1 + 2 X2 <= 120; X1, X2>=0\ \>", "Text", CellChangeTimes->{{3.497933948401892*^9, 3.497933963644764*^9}, { 3.497933997996729*^9, 3.4979340539579296`*^9}, 3.497936872739773*^9, 3.5263711891786127`*^9, 3.5330132173946743`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NMaximize", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"20", " ", "x1"}], "+", RowBox[{"30", " ", "x2"}]}], ",", RowBox[{"x1", "\[LessEqual]", " ", "60"}], ",", RowBox[{"x2", "\[LessEqual]", " ", "50"}], ",", RowBox[{ RowBox[{"x1", "+", RowBox[{"2", " ", "x2"}]}], "\[LessEqual]", "120"}], ",", RowBox[{"x1", "\[GreaterEqual]", "0"}], ",", RowBox[{"x2", "\[GreaterEqual]", "0"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"x1", ",", "x2"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.5330131075050807`*^9, 3.533013180469409*^9}, { 3.533013227924692*^9, 3.5330133111196384`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"2100.`", ",", RowBox[{"{", RowBox[{ RowBox[{"x1", "\[Rule]", "60.`"}], ",", RowBox[{"x2", "\[Rule]", "30.`"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.533013315300446*^9, 3.533013623422394*^9, 3.5330139099289017`*^9, 3.5330139552503824`*^9, 3.5330149767695007`*^9}] }, Open ]], Cell["\<\ b) \[DownQuestion]Se consumen todas las horas de trabajo disponibles? Si; \ \[DownQuestion]Se agota la capacidad de producci\[OAcute]n de las dos l\ \[IAcute]neas de producci\[OAcute]n? No, en la l\[IAcute]nea 2 hay una \ holgura de 20 pues tiene una capacidad de 50 y consumimos 30. \ \>", "Text", CellChangeTimes->{{3.497933948401892*^9, 3.497933963644764*^9}, { 3.497933997996729*^9, 3.4979340539579296`*^9}, 3.497936872739773*^9, 3.5263711891786127`*^9, 3.5330132173946743`*^9, {3.5330133454708986`*^9, 3.5330133601817245`*^9}}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Ejercicio 3.- Evacuaci\[OAcute]n", "Subsection", CellChangeTimes->{{3.438519533890625*^9, 3.438519539390625*^9}, { 3.438519577078125*^9, 3.438519579796875*^9}, {3.4652968547758355`*^9, 3.465296870641035*^9}, 3.494334773008932*^9, {3.4943358082511444`*^9, 3.4943358087631736`*^9}, {3.4979367645319653`*^9, 3.4979367651239996`*^9}}], Cell["\<\ Una cat\[AAcute]strofe obliga a evacuar tres nucleos de poblaci\[OAcute]n \ (Salamanca, Avila y Zamora) hacia dos campamentos de refugiados (A y B). La \ poblaci\[OAcute]n a evacuar de cada ciudad y las capacidades m\[AAcute]ximas \ de los campamentos se muestran en la tabla. La duraci\[OAcute]n, en horas, de \ cada trayecto tambien se muestra en la tabla.\ \>", "Text", CellChangeTimes->{{3.4385195879375*^9, 3.43851991409375*^9}, { 3.438520037109375*^9, 3.4385202459375*^9}, {3.43852036365625*^9, 3.43852039121875*^9}, {3.43852042571875*^9, 3.438520563296875*^9}, { 3.438520596765625*^9, 3.4385206145625*^9}, {3.438520679375*^9, 3.43852069690625*^9}, {3.438530566921875*^9, 3.438530570421875*^9}, { 3.4385306305625*^9, 3.438530633734375*^9}}], Cell[BoxData[GridBox[{ {"Campamento", RowBox[{"Capacidad", " ", "m\[AAcute]xima"}], "Salamanca", "Avila", "Zamora"}, {"A", "310000", "8", "6", "4"}, {"B", "260000", "5", "4", "6"}, {"\[Placeholder]", RowBox[{ RowBox[{"Ciudad", RowBox[{"(", "poblaci\[OAcute]n", ")"}]}], "\[Rule]", " "}], "150000", "70000", "90000"} }, GridBoxDividers->{ "Columns" -> {{True}}, "ColumnsIndexed" -> {}, "Rows" -> {{True}}, "RowsIndexed" -> {}}]], "Text", CellChangeTimes->{{3.43851992475*^9, 3.43852002353125*^9}, { 3.438520071640625*^9, 3.43852008653125*^9}, {3.43852025*^9, 3.4385203350625*^9}, {3.438520642375*^9, 3.438520644875*^9}}], Cell["\<\ Se pide: minimizar el tiempo total de evacuaci\[OAcute]n de la poblaci\ \[OAcute]n desde las ciudades a los campamentos de refugiados, sin exceder \ las capacidades estos. \ \>", "Text", CellChangeTimes->{{3.438520702875*^9, 3.438520756375*^9}, { 3.43852094609375*^9, 3.43852096728125*^9}, {3.438530580828125*^9, 3.43853060453125*^9}}], Cell["\<\ a) Plantee matem\[AAcute]ticamente el problema (funci\[OAcute]n objetivo y \ restricciones) \ \>", "Text", CellChangeTimes->{{3.438520975125*^9, 3.438521052875*^9}, { 3.43853061653125*^9, 3.43853062046875*^9}}], Cell[TextData[{ "Defina las variables como sigue: ", Cell[BoxData[ FormBox[ SubscriptBox["x", "ij"], TraditionalForm]]], " donde ", Cell[BoxData[ FormBox["i", TraditionalForm]]], ":{1, 2, 3} indica la poblaci\[OAcute]n de origen {1 = Salamanca, 2 =Avila, \ 3 = Zamora} y ", Cell[BoxData[ FormBox["j", TraditionalForm]]], ": {1, 2} indica el destino {1=Campamento A, 2 = Campamento B}. " }], "Text", CellChangeTimes->{{3.438520765359375*^9, 3.43852084521875*^9}}], Cell["\<\ b) Indique cual es el tiempo total en horas que minimiza el trasporte\ \>", "Text", CellChangeTimes->{{3.43852101734375*^9, 3.438521019*^9}, {3.4385211204375*^9, 3.438521168234375*^9}}], Cell["\<\ c) Complete la siguiente tabla con el n\[UAcute]mero de personas que enviaria \ desde ciudades a campamentos\ \>", "Text", CellChangeTimes->{{3.43852101734375*^9, 3.438521019*^9}, {3.4385211204375*^9, 3.4385212116875*^9}, {3.4385212735*^9, 3.4385213401875*^9}, { 3.4653037670826*^9, 3.4653037677066*^9}, {3.526642428008156*^9, 3.526642429053358*^9}, 3.526642496492276*^9}], Cell[BoxData[GridBox[{ {"Campamento", "Salamanca", "Avila", "Zamora"}, {"A", StyleBox["\[Placeholder]", FontColor->GrayLevel[1]], StyleBox["\[Placeholder]", FontColor->GrayLevel[1]], StyleBox["\[Placeholder]", FontColor->GrayLevel[1]]}, {"B", StyleBox["\[Placeholder]", FontColor->GrayLevel[1]], StyleBox["\[Placeholder]", 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misma restriccion):\ \>", "Text", CellChangeTimes->{{3.4954672190590544`*^9, 3.4954672233178616`*^9}, { 3.4954672574975214`*^9, 3.4954673220192347`*^9}, {3.495467365917712*^9, 3.495467437131837*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"r6", "=", " ", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"#", "\[GreaterEqual]", " ", "0"}], ")"}], "&"}], ",", " ", "var"}], "]"}]}]], "Input", CellChangeTimes->{{3.4954672471235037`*^9, 3.4954672529423137`*^9}, { 3.495467325466841*^9, 3.495467361346904*^9}, {3.4954674569594717`*^9, 3.495467458020274*^9}, {3.4979391937656536`*^9, 3.497939195085655*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["x", "11"], "\[GreaterEqual]", "0"}], ",", RowBox[{ SubscriptBox["x", "12"], "\[GreaterEqual]", "0"}], ",", RowBox[{ SubscriptBox["x", "21"], "\[GreaterEqual]", "0"}], ",", RowBox[{ SubscriptBox["x", "22"], "\[GreaterEqual]", "0"}], ",", RowBox[{ SubscriptBox["x", "31"], "\[GreaterEqual]", "0"}], ",", RowBox[{ 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Para ello extraemos el segundo \ termino de la solucion (que es el resultado de cada una de las variables) y \ lo sustituimos empleando /. que significa \"reemplaza por\"\ \>", "Text", CellChangeTimes->{{3.497939349389957*^9, 3.4979394224060693`*^9}, 3.497939512608203*^9, {3.497939547844265*^9, 3.4979395563042774`*^9}, { 3.4979396347663946`*^9, 3.4979397449145536`*^9}, {3.49794031730949*^9, 3.497940339863527*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{"desplazados", " ", "=", RowBox[{"sol", "[", RowBox[{"[", "2", "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.497939760548578*^9, 3.497939779414611*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["x", "11"], "\[Rule]", "0.`"}], ",", RowBox[{ SubscriptBox["x", "12"], "\[Rule]", "150000.`"}], ",", RowBox[{ SubscriptBox["x", "21"], "\[Rule]", "0.`"}], ",", RowBox[{ SubscriptBox["x", "22"], "\[Rule]", "70000.`"}], ",", RowBox[{ SubscriptBox["x", "31"], "\[Rule]", "90000.`"}], ",", RowBox[{ SubscriptBox["x", "32"], "\[Rule]", "0.`"}]}], "}"}]], "Output", CellChangeTimes->{ 3.4979397746726*^9, {3.4979398704607487`*^9, 3.4979398792097864`*^9}, 3.4979403912167425`*^9, 3.4979410100713773`*^9, 3.5330136240307946`*^9, 3.5330139107089033`*^9, 3.5330139560615835`*^9, 3.533014977440302*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"totaldesplazados", "=", " ", RowBox[{ RowBox[{ SubscriptBox["x", "11"], "+", " ", SubscriptBox["x", "12"], "+", SubscriptBox["x", "21"], "+", " ", SubscriptBox["x", "22"], "+", " ", SubscriptBox["x", "31"], "+", " ", SubscriptBox["x", "32"]}], "/.", " ", RowBox[{"sol", "[", RowBox[{"[", "2", "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.4979395911023307`*^9, 3.4979396328763914`*^9}, 3.4979397556965694`*^9, {3.4979398044086485`*^9, 3.497939806658652*^9}}], Cell[BoxData["310000.`"], "Output", CellChangeTimes->{{3.4979398705697546`*^9, 3.497939879260789*^9}, 3.4979403912587447`*^9, 3.49794101011738*^9, 3.533013624249195*^9, 3.5330139108181033`*^9, 3.533013956186384*^9, 3.533014977674302*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"tiempomedio", " ", "=", " ", RowBox[{"totalhoras", "/", "totaldesplazados"}]}]], "Input", CellChangeTimes->{{3.4979398088786545`*^9, 3.4979398336486964`*^9}, { 3.497940270479416*^9, 3.497940277801427*^9}}], Cell[BoxData["4.483870967741935`"], "Output", CellChangeTimes->{{3.497939870612757*^9, 3.4979398793067923`*^9}, 3.497940278431428*^9, 3.497940391363749*^9, 3.4979410101513815`*^9, 3.533013624358395*^9, 3.5330139109273033`*^9, 3.533013956311184*^9, 3.533014977799102*^9}] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Ejercicio 4.- Problema de transporte ", "Subsection", CellChangeTimes->{{3.4952794015443964`*^9, 3.495279407987208*^9}, { 3.4954666470996494`*^9, 3.495466650188455*^9}, {3.497934240758614*^9, 3.497934248364049*^9}, {3.497936825059697*^9, 3.4979368256996975`*^9}, { 3.4979368823997865`*^9, 3.4979369288038974`*^9}, {3.526371224699875*^9, 3.5263712391143003`*^9}, {3.533054211290755*^9, 3.533054237982402*^9}}], Cell["\<\ La cadena de grandes almacenes CARREFA desea contratar el suministro de cajas \ de leche para los centros que tiene en Barcelona, Madrid, Sevilla y \ Valladolid. Para ello recibe oferta de tres envasadores: Leche Pascualo, \ Leche Simone y Leche Pulova. Los precios son los siguientes: Pascualo 3.00 \ \[Euro] por caja, Simone 2.80 \[Euro] por caja y Pulova 2.70 \[Euro] por \ caja. El m\[AAcute]ximo n\[UAcute]mero de cajas que diariamente puede \ suministrar cada envasador se indica en la tabla adjunta, en la columna \ \"Suministro m\[AAcute]ximo\". La demanda diaria de cada centro se muestra \ en la fila \"Demanda\". Los costes de transporte, en \[Euro] por caja, se \ indican en la misma tabla. \ \>", "Text"], Cell[TextData[{ Cell[BoxData[ StyleBox[GridBox[{ {"\[Placeholder]", RowBox[{"Suministro", " ", "m\[AAcute]ximo"}], "Barcelona", "Madrid", "Sevilla", "Valladolid"}, {"Pascualo", "2500", RowBox[{"0.5", " ", "\[Euro]"}], RowBox[{"0.6", "\[Euro]"}], RowBox[{"0.7", " ", "\[Euro]"}], RowBox[{"0.5", "\[Euro]"}]}, {"Simone", "1700", RowBox[{"0.5", "\[Euro]"}], RowBox[{"0.6", "\[Euro]"}], RowBox[{"0.8", "\[Euro]"}], RowBox[{"0.5", "\[Euro]"}]}, {"Pulova", "1500", RowBox[{"0.4", "\[Euro]"}], RowBox[{"0.5", "\[Euro]"}], RowBox[{"0.9", " ", "\[Euro]"}], RowBox[{"0.7", "\[Euro]"}]}, {"\[Placeholder]", RowBox[{"Demanda", "\[Rule]"}], "800", "1100", "500", "300"} }, GridBoxDividers->{ "Columns" -> {{True}}, "ColumnsIndexed" -> {}, "Rows" -> {{True}}, "RowsIndexed" -> {}}], FontSize->10]]], "\n" }], "Example", TextAlignment->Center], Cell["\<\ CARREFA desea calcular las cantidades que debe encargar a cada envasador para \ minimizar el coste total.\ \>", "Text"], Cell[CellGroupData[{ Cell["a) Formular matematicamente el problema ", "Subsubsection"], Cell[TextData[{ "Sugerencia: Las variables se pueden definir como sigue: ", Cell[BoxData[ FormBox[ SubscriptBox["x", "ij"], TraditionalForm]]], " = n\[UAcute]mero de cajas del envasador ", StyleBox["i", FontSlant->"Italic"], " , ", Cell[BoxData[ FormBox["i", TraditionalForm]]], ":{1, 2, 3}, siendo {1 = Pascualo, 2 = Simone, 3 = Pulova}, con destino a ", Cell[BoxData[ FormBox["j", TraditionalForm]]], ": {1, 2, 3, 4} , siendo {1= Barcelona, 2 = Madrid, 3 = Sevilla, 4 = \ Valladolid}. " }], "Text"], Cell["\<\ Nota: Tenga en cuenta que el coste total de cada caja est\[AAcute] dado por \ el precio de compra m\[AAcute]s el de transporte.\ \>", "Text"], Cell["Nos aseguramos de borrar todas las asignaciones previas.", "Text", CellChangeTimes->{{3.4954667841926904`*^9, 3.4954668044571257`*^9}, { 3.495466878541656*^9, 3.495466888291673*^9}, {3.4954670721847963`*^9, 3.4954670785652075`*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.466151159341*^9, 3.46615116226*^9}, 3.466151198587*^9}], Cell["Se definen las variables", "Text", CellChangeTimes->{{3.495466856405217*^9, 3.4954668647044315`*^9}, { 3.5263714186706157`*^9, 3.5263714273286304`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"var", "=", RowBox[{"{", RowBox[{ SubscriptBox["x", "11"], ",", SubscriptBox["x", "12"], ",", SubscriptBox["x", "13"], ",", SubscriptBox["x", "14"], ",", SubscriptBox["x", "21"], ",", SubscriptBox["x", "22"], ",", SubscriptBox["x", "23"], ",", SubscriptBox["x", 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precio m\[AAcute]ximo deber\[IAcute]a vender Leche \ Pascualo para saturar su capacidad de envasado?\ \>", "Subsubsection", CellChangeTimes->{{3.497704228887779*^9, 3.4977042335660467`*^9}}], Cell["\<\ Las restricciones y las variables son las mismas independientemente del \ precio de la leche, luego no hay que escribirlas de nuevo\ \>", "Text", CellChangeTimes->{{3.495466544482669*^9, 3.495466585042741*^9}, { 3.495472690147133*^9, 3.495472708633165*^9}}], Cell["\<\ Los precios de compra iniciales son Pascualo 3.00 \[Euro] , Simone 2.80 \ \[Euro] y Pulova 2.70 \[Euro], en vez de escribirlos directamente en la funci\ \[OAcute]n de optimizaci\[OAcute]n lo definimos previamente {pa, sim, pul} = \ {3, 2.8, 2.7}; vamos ejecutando sucesivamente la celda bajando el precio (pa) \ hasta que se obtenga que Pascualo vende 2500 uds. \ \>", "Text", CellChangeTimes->{{3.4676362215806*^9, 3.4676362756814003`*^9}, { 3.4952799019932756`*^9, 3.49527991021449*^9}, {3.4979407338307886`*^9, 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anterior comparando notaci\[OAcute]n utilizada \ on NMinimize con la de LinearProgramming (LP). En la pr\[AAcute]ctica esto no \ es necesario, de hecho es m\[AAcute]s sencillo realizarlo directamente \ utilzando la natacion de LP, aqui lo realizamos a efectos did\[AAcute]cticos \ \>", "Text", CellChangeTimes->{{3.495509650580809*^9, 3.495509673436116*^9}, { 3.495509739987389*^9, 3.4955097531161404`*^9}, {3.4955098314170485`*^9, 3.495509979772534*^9}, {3.495511085523779*^9, 3.4955111761149607`*^9}, { 3.495516149780859*^9, 3.4955161855579057`*^9}}], Cell[TextData[{ "Con LinearProgramming (LP) no es necesario definir explicitamente las \ variables. \n", Cell[BoxData[ RowBox[{"{", RowBox[{ SubscriptBox["x", "11"], ",", SubscriptBox["x", "12"], ",", SubscriptBox["x", "13"], ",", SubscriptBox["x", "14"], ",", SubscriptBox["x", "21"], ",", SubscriptBox["x", "22"], ",", SubscriptBox["x", "23"], ",", SubscriptBox["x", "24"], ",", SubscriptBox["x", "31"], ",", SubscriptBox["x", "32"], ",", SubscriptBox["x", "33"], ",", SubscriptBox["x", "34"]}], "}"}]], CellChangeTimes->{{3.495466847216801*^9, 3.4954668525052104`*^9}}], " equivale a {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}. " }], "Text", CellChangeTimes->{{3.4955100099962626`*^9, 3.495510055792882*^9}, { 3.495510109629961*^9, 3.4955104459892*^9}, {3.4955104829853163`*^9, 3.4955105281619*^9}, 3.4955111852854853`*^9, {3.4955113104756455`*^9, 3.4955113168440104`*^9}, 3.526371836049349*^9}], Cell["La funci\[OAcute]n a optimizar es fo", "Text", CellChangeTimes->{{3.495509218993124*^9, 3.495509227523612*^9}, { 3.4955161261685085`*^9, 3.4955161295777035`*^9}, {3.5263718453313646`*^9, 3.5263718477961693`*^9}}], Cell[TextData[{ "Para construir la funci\[OAcute]n de optimizaci\[OAcute]n hemos de utilizar \ los terminos agrupados que multiplican a cada variable. Por ejemplo: el coste \ total de ", Cell[BoxData[ SubscriptBox["x", "11"]], CellChangeTimes->{{3.4676352960014*^9, 3.4676353040354*^9}, { 3.4676361640945997`*^9, 3.4676362037497997`*^9}, {3.4954730403521476`*^9, 3.495473041069749*^9}}], " unidades ser\[AAcute] el coste de trasporte + el de compra, esto es (0.5 + \ pa) y as\[IAcute] para todas las variables" }], "Text", CellChangeTimes->{{3.495513707934255*^9, 3.495513907731683*^9}, { 3.495516058507639*^9, 3.4955160659990673`*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.466151159341*^9, 3.46615116226*^9}, 3.466151198587*^9}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"pa", ",", " ", "sim", ",", " ", "pul"}], "}"}], " ", "=", RowBox[{"{", RowBox[{"3", ",", " ", "2.8", ",", "2.7"}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.467635645691*^9, 3.4676356502462*^9}, { 3.4676360916794*^9, 3.4676361599918003`*^9}, {3.495473036545741*^9, 3.4954730372321424`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"fo", "=", RowBox[{"{", RowBox[{ RowBox[{"(", RowBox[{"0.5", "+", "pa"}], ")"}], ",", RowBox[{"(", RowBox[{"0.6", "+", "pa"}], ")"}], ",", " ", RowBox[{"(", RowBox[{"0.7", "+", "pa"}], ")"}], ",", RowBox[{"(", RowBox[{"0.5", " ", "+", "pa"}], ")"}], ",", " ", RowBox[{"(", RowBox[{"0.5", "+", "sim"}], ")"}], ",", RowBox[{"(", RowBox[{"0.6", "+", "sim"}], ")"}], ",", RowBox[{"(", RowBox[{"0.8", "+", "sim"}], ")"}], ",", RowBox[{"(", RowBox[{"0.5", " ", "+", "sim"}], ")"}], ",", " ", RowBox[{"(", RowBox[{"0.4", "+", "pul"}], ")"}], ",", RowBox[{"(", RowBox[{"0.5", "+", "pul"}], ")"}], ",", RowBox[{"(", RowBox[{"0.9", "+", "pul"}], ")"}], ",", RowBox[{"(", RowBox[{"0.7", "+", "pul"}], ")"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.495509145282908*^9, 3.495509167491178*^9}, { 3.4955092335979595`*^9, 3.4955092361701064`*^9}}], Cell["Restricciones", "Text", CellChangeTimes->{{3.4955095499480534`*^9, 3.4955095530112286`*^9}}], Cell[TextData[{ "La restricci\[OAcute]n ", Cell[BoxData[ RowBox[{ RowBox[{"r1", "=", RowBox[{ RowBox[{ SubscriptBox["x", "11"], "+", SubscriptBox["x", "12"], "+", SubscriptBox["x", "13"], "+", SubscriptBox["x", "14"]}], "\[LessEqual]", "2500"}]}], ";"}]], CellChangeTimes->{{3.4676353462646*^9, 3.4676354057318*^9}, 3.4955093723908978`*^9, {3.4955094318853006`*^9, 3.4955094502203493`*^9}}], " se escribir\[AAcute] (cuando falta alguna de las variables se entiende que \ esta toma valor 0 y se escribe 0 en la posici\[OAcute]n correspondiente)" }], "Text", CellChangeTimes->{{3.4955100099962626`*^9, 3.495510055792882*^9}, { 3.495510109629961*^9, 3.4955104459892*^9}, {3.4955104829853163`*^9, 3.4955105281619*^9}, 3.4955111852854853`*^9, {3.495516076814686*^9, 3.4955160775977306`*^9}, {3.4955162053660383`*^9, 3.495516228422357*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"m1", "=", " ", RowBox[{"{", RowBox[{ "1", ",", "1", ",", "1", ",", "1", ",", " ", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}]}], ";", " ", RowBox[{"b1", " ", "=", RowBox[{"{", RowBox[{"2500", ",", RowBox[{"-", "1"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.4955092428184867`*^9, 3.4955092577463408`*^9}, { 3.4955094575297675`*^9, 3.4955095302359257`*^9}, {3.495509600132924*^9, 3.495509622630211*^9}, {3.4955106084224906`*^9, 3.495510608654504*^9}}], Cell["\<\ donde el -1 equivale a \"\[LessEqual]\"; 0 a \"==\"; y 1 a \"\[GreaterEqual]\ \".\ \>", "Text", CellChangeTimes->{{3.4955104846614122`*^9, 3.49551049110678*^9}, 3.495510561848827*^9}], Cell[TextData[{ " La restricci\[OAcute]n ", Cell[BoxData[ RowBox[{"r2", "=", RowBox[{ RowBox[{ SubscriptBox["x", "21"], "+", SubscriptBox["x", "22"], "+", SubscriptBox["x", "23"], "+", SubscriptBox["x", "24"]}], "\[LessEqual]", "1700"}]}]], CellChangeTimes->{{3.4676353462646*^9, 3.4676354057318*^9}, 3.4955093723908978`*^9, {3.4955094318853006`*^9, 3.4955094502203493`*^9}, 3.4955105395865536`*^9}], " se escribe" }], "Text", CellChangeTimes->{ 3.495510578025752*^9, 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Si \ queremos calcular el valor del punto \[OAcute]ptimo lo que tenemos que hacer \ es multiplicar la funci\[OAcute]n a optimizar por la soluci\[OAcute]n \ obtenida (utilizamos \".\" para multiplicar matrices)." }], "Text", CellChangeTimes->{{3.495512208542495*^9, 3.495512309282257*^9}, { 3.4955160184993505`*^9, 3.4955160192113914`*^9}, {3.495516274747007*^9, 3.4955163397937274`*^9}, {3.526642494214672*^9, 3.526642530890337*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"fo", ".", "sol"}]], "Input", CellChangeTimes->{{3.495512176480661*^9, 3.495512182731018*^9}}], Cell[BoxData["8870.`"], "Output", CellChangeTimes->{{3.495513455604823*^9, 3.495513495692116*^9}, 3.497934561105937*^9, 3.497940404057234*^9, 3.497941023435763*^9, 3.5330136278996015`*^9, 3.533013792015894*^9, 3.5330138228123484`*^9, 3.533013912940707*^9, 3.5330139581363873`*^9, 3.533014980513507*^9}] }, Open ]], Cell["\<\ Las restricciones y las variables son las mismas independientemente del \ precio de la leche, luego no hay que escribirlas de nuevo\ \>", "Text", CellChangeTimes->{{3.495466544482669*^9, 3.495466585042741*^9}, { 3.495472690147133*^9, 3.495472708633165*^9}}], Cell["\<\ Los precios de venta son: Pascualo ? \[Euro] , Simone 2.80 \[Euro]y Pulova \ 2.70 \[Euro] que puedo escribirlo como sigue\ \>", "Text", CellChangeTimes->{{3.4676362215806*^9, 3.4676362756814003`*^9}, { 3.4952799019932756`*^9, 3.49527991021449*^9}}], Cell["\<\ Pruebe a ir bajando el precio (pa) hasta que obtenga que Pascualo vende 2500 \ uds. Podemos empezar por 3 \[Euro] y ir reduciendo 0.1 \[Euro] hasta que \ obtenemos 2500 \[Euro]\ \>", "Text", CellChangeTimes->{{3.4676363809658003`*^9, 3.4676364761414003`*^9}, { 3.49547290503751*^9, 3.4954729052559104`*^9}, {3.495472970963226*^9, 3.4954730239253187`*^9}, {3.495513230873969*^9, 3.4955132341701574`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"pa", ",", " ", "sim", ",", " ", "pul"}], "}"}], " ", "=", RowBox[{"{", RowBox[{"2.6", ",", " ", "2.8", ",", "2.7"}], "}"}]}], ";"}], "\n", RowBox[{ RowBox[{"fo", "=", RowBox[{"{", RowBox[{ RowBox[{"(", RowBox[{"0.5", "+", "pa"}], ")"}], ",", RowBox[{"(", RowBox[{"0.6", "+", "pa"}], ")"}], ",", " ", RowBox[{"(", RowBox[{"0.7", "+", "pa"}], ")"}], ",", RowBox[{"(", RowBox[{"0.5", " ", "+", "pa"}], ")"}], ",", " ", RowBox[{"(", RowBox[{"0.5", "+", "sim"}], ")"}], ",", RowBox[{"(", RowBox[{"0.6", "+", "sim"}], ")"}], ",", RowBox[{"(", RowBox[{"0.8", "+", "sim"}], ")"}], ",", RowBox[{"(", RowBox[{"0.5", " ", "+", "sim"}], ")"}], ",", " ", RowBox[{"(", RowBox[{"0.4", "+", "pul"}], ")"}], ",", RowBox[{"(", RowBox[{"0.5", "+", "pul"}], ")"}], ",", RowBox[{"(", RowBox[{"0.9", "+", "pul"}], ")"}], ",", RowBox[{"(", RowBox[{"0.7", "+", "pul"}], ")"}]}], "}"}]}], ";"}], "\n", RowBox[{ RowBox[{"sol", "=", RowBox[{"LinearProgramming", "[", RowBox[{"fo", ",", RowBox[{"{", RowBox[{ "m1", ",", "m2", ",", "m3", ",", "m4", ",", "m5", ",", "m6", ",", "m7"}], "}"}], ",", RowBox[{"{", RowBox[{ "b1", ",", "b2", ",", "b3", ",", "b4", ",", "b5", ",", "b6", ",", " ", "b7"}], "}"}]}], "]"}]}], ";"}], "\n", RowBox[{"Total", "[", RowBox[{"Take", "[", RowBox[{"sol", ",", " ", RowBox[{"{", RowBox[{"1", ",", "4"}], "}"}]}], "]"}], "]"}]}], "Input", CellChangeTimes->{{3.467635645691*^9, 3.4676356502462*^9}, { 3.4676360916794*^9, 3.4676361599918003`*^9}, {3.495473036545741*^9, 3.4954730372321424`*^9}, {3.495513201661298*^9, 3.495513215244075*^9}}], Cell[BoxData["2500.`"], "Output", CellChangeTimes->{{3.495513455927841*^9, 3.495513496106139*^9}, 3.4979345613299494`*^9, 3.497940404279247*^9, 3.497941024131773*^9, 3.533013629132004*^9, 3.5330137921874943`*^9, {3.533013823015149*^9, 3.5330138352611704`*^9}, 3.533013913049907*^9, 3.5330139581987877`*^9, 3.5330149805759068`*^9}] }, Open ]] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Ejemplos de aplicaciones de optimizaci\[OAcute]n no lineal\ \>", "Section", CellChangeTimes->{ 3.525255611161892*^9, 3.5252556466479216`*^9, 3.5252556911514673`*^9, { 3.5252606776502204`*^9, 3.525260680386377*^9}, {3.525333093943085*^9, 3.525333094537119*^9}, {3.5254570457656236`*^9, 3.5254570513504333`*^9}, 3.562827663813861*^9}], Cell[BoxData[ RowBox[{"Clear", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.466151159341*^9, 3.46615116226*^9}, 3.466151198587*^9}], Cell[CellGroupData[{ Cell["Ejemplo 1.- Funciones Cobb-Douglas", "Subsection", CellChangeTimes->{{3.5263708600492344`*^9, 3.5263708719208555`*^9}, { 3.5330542824112797`*^9, 3.533054286966488*^9}}], Cell[TextData[{ "En muchos problemas econ\[OAcute]micos aparece funciones del tipo \ (funciones Cobb-Douglas): ", Cell[BoxData[ FormBox[ RowBox[{"F", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}]}], TraditionalForm]]], " = A ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "a"], SuperscriptBox["y", "b"]}], TraditionalForm]]], " con ", Cell[BoxData[ FormBox[ RowBox[{"x", ",", " ", RowBox[{"y", " ", "\[GreaterEqual]", " ", "0"}]}], TraditionalForm]]], " y A, a, b ctes. Por ejemplo: " }], "Text", CellChangeTimes->{{3.465360930557598*^9, 3.465361134387198*^9}, { 3.465361233899598*^9, 3.465361240295598*^9}, {3.4653613345663977`*^9, 3.4653613437079983`*^9}, {3.465361534651998*^9, 3.465361551141198*^9}, { 3.465361677429598*^9, 3.465361677812598*^9}, {3.465361786590598*^9, 3.465361888764598*^9}, {3.465362032929598*^9, 3.465362047440598*^9}, 3.465737233677072*^9, {3.4661781121472273`*^9, 3.466178126078027*^9}, { 3.494334976404565*^9, 3.4943349824679117`*^9}, {3.5252607443800373`*^9, 3.5252608140930243`*^9}}], Cell[TextData[{ "El consumo de leche en cierta regi\[OAcute]n sigue la funci\[OAcute]n ", Cell[BoxData[ FormBox[ RowBox[{"c", "(", RowBox[{"p", ",", " ", "r"}], ")"}], TraditionalForm]]], " = 1.3 ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["p", RowBox[{"-", "1.5"}]], SuperscriptBox["r", "0.01"]}], TraditionalForm]]], " donde ", Cell[BoxData[ FormBox["p", TraditionalForm]]], " es el precio del L de leche y ", Cell[BoxData[ FormBox["r", TraditionalForm]]], " renta diaria por familia en \[Euro]/d\[IAcute]a. El rango de validez de la \ expresi\[OAcute]n anterior es para las rentas familiares comprendidas entre \ 20 y 200 \[Euro]/dia. El precio del L leche oscila entre 0.4 \[Euro] y 1.2 \ \[Euro] \[DownQuestion]Cual ser\[AAcute] el consumo m\[AAcute]ximo y m\ \[IAcute]nimo? 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centrales el\[EAcute]ctricas (una \ nuclear, dos de gas y una de carb\[OAcute]n) cuyo beneficio por hora de \ funcionamiento depende del n\[UAcute]mero de horas que funcione al \ a\[NTilde]o. Estos valores se muestran en la tabla donde t1, t2, t3, t4 \ indica las horas de funcionamiento de la central correspondiente. Cada \ central debe funcionar al a\[NTilde]o al menos 5000 h si es nuclear , 2000 h \ si es de carb\[OAcute]n, y 2500 h si es de gas. Ademas el conjunto de las \ centrales pueden llegar a funcionar conjuntamente un m\[AAcute]ximo de 20000 \ horas. Considerese un a\[NTilde]o normal, es decir de 8760 h. \[DownQuestion]Cual es el beneficio anual esperado? \[DownQuestion]Cuantas \ horas debe funcionar cada central para que el beneficio anual se \ m\[AAcute]ximo \[DownQuestion]Cual sera este beneficio?\ \>", "Text", CellChangeTimes->{{3.4993301626866355`*^9, 3.499330213558325*^9}, { 3.499330249391588*^9, 3.4993303128992996`*^9}, {3.499330345191356*^9, 3.499330414252678*^9}, {3.499330457807954*^9, 3.4993305196308627`*^9}, { 3.4993305524845204`*^9, 3.4993306124822254`*^9}, {3.499330701043581*^9, 3.4993307018235826`*^9}, {3.499331123351923*^9, 3.499331324639076*^9}, { 3.4993486939314923`*^9, 3.4993487362231665`*^9}, {3.4993488027260838`*^9, 3.499348811524499*^9}, {3.499348853192172*^9, 3.4993489754183865`*^9}, { 3.4993495536802025`*^9, 3.4993496187791166`*^9}, {3.4993498646823483`*^9, 3.4993498824351797`*^9}, {3.4993500922555485`*^9, 3.499350097668758*^9}, { 3.49935018429571*^9, 3.499350192891325*^9}, {3.499350279377877*^9, 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Durante ese periodo hay que ir \ traspasando el dinero entre tres fondos A, B y C. Cada uno de los fondos da \ una rentabilidad que se c\[AAcute]lcula seg\[UAcute]n el tiempo que se esta \ en ese fondo (Ver tabla; ta, tb y tc son los tiempos de permanencia, en a\ \[NTilde]os, en el fondo A, B y C respectivamente y Co es el capital inicial \ que se invierte) con un tiempo m\[IAcute]nimo de permancia en cada fondo de \ un a\[NTilde]o. En el tercer fondo hay que mantener la inversi\[OAcute]n un \ tiempo superior a la suma de los dos primeros periodos. La rentabilidad total \ al final del periodo de 7 a\[NTilde]os ser\[AAcute] la suma de las \ rentabilidades de cada intervalo . Dispongo de 100 000 \[Euro]. \ \[DownQuestion]Que tiempos de permancia en cada fondo debo elegir de forma \ que la rentabilidad acumulada en los 7 a\[NTilde]os sea m\[AAcute]xima?\ \>", "Text", CellChangeTimes->{{3.4979350449576116`*^9, 3.497935056686282*^9}, 3.5263722614932957`*^9}], Cell[TextData[{ "\n", Cell[BoxData[GridBox[{ { StyleBox["Fondo", FontSize->10], StyleBox["Rentabilidad", FontSize->10], StyleBox[ RowBox[{"Tiempo", " ", "M\[IAcute]nimo"}], FontSize->10], StyleBox[ RowBox[{"Tiempo", " ", "Elegido"}], FontSize->10]}, { StyleBox["A", FontSize->10], StyleBox[ RowBox[{"0.5", "*", RowBox[{"ta", "^", "1.1"}]}], FontSize->10], StyleBox["1", FontSize->10], StyleBox["\[Placeholder]", FontSize->10]}, { StyleBox["B", FontSize->10], StyleBox[ RowBox[{"0.5", "*", RowBox[{"E", "^", RowBox[{"(", RowBox[{"0.2", "tb"}], ")"}]}]}], FontSize->10], StyleBox["1", FontSize->10], StyleBox["\[Placeholder]", FontSize->10]}, { StyleBox["C", FontSize->10], StyleBox[ RowBox[{"0.3", "*", 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Se admite un rango en la temperatura interior, ", StyleBox["Ti,", FontSlant->"Italic"], " entre 20 y 23 grados y de la humedad interior, ", StyleBox["hi, ", FontSlant->"Italic"], "entre 0.5 y 0.6. 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\ \[DownQuestion]cual ser\[AAcute] el consumo por hora?" }], "Text", CellChangeTimes->{{3.465360930557598*^9, 3.465361134387198*^9}, { 3.465361233899598*^9, 3.465361240295598*^9}, {3.4653613345663977`*^9, 3.4653613437079983`*^9}, {3.465361534651998*^9, 3.465361551141198*^9}, { 3.465361677429598*^9, 3.465361677812598*^9}, {3.465361786590598*^9, 3.465361888764598*^9}, {3.4653619228385983`*^9, 3.465362014282598*^9}, { 3.465362078256598*^9, 3.465362243948598*^9}, {3.465362431517598*^9, 3.4653624323575983`*^9}, {3.465362647110598*^9, 3.465362695911598*^9}, { 3.4653627378175983`*^9, 3.465362753759598*^9}, {3.465362843018598*^9, 3.465362851002598*^9}, 3.465363018646598*^9, {3.465363082418598*^9, 3.4653630900495977`*^9}, {3.465363130429598*^9, 3.465363167829598*^9}, { 3.465363213203598*^9, 3.465363217297598*^9}, {3.4653634059989977`*^9, 3.465363511954198*^9}, {3.465363943980598*^9, 3.4653639693929977`*^9}, { 3.465364157731798*^9, 3.465364203330598*^9}, {3.465364773307798*^9, 3.4653647744153976`*^9}, {3.465364825240198*^9, 3.4653648382349977`*^9}, { 3.465364944923398*^9, 3.465365064840598*^9}, {3.465365111000998*^9, 3.465365114058598*^9}, {3.465365221121398*^9, 3.4653653426609983`*^9}, { 3.4657372507122717`*^9, 3.465737251117872*^9}, 3.466151582539*^9, { 3.466178137356827*^9, 3.4661781440180273`*^9}, {3.469697408124799*^9, 3.4696974167983985`*^9}, {3.4697119185579867`*^9, 3.469712183960787*^9}, { 3.469712219762787*^9, 3.469712222196387*^9}, {3.4697150941596003`*^9, 3.4697154653928003`*^9}, {3.4697155455768003`*^9, 3.4697155486656*^9}, { 3.4697156331084003`*^9, 3.4697156754312*^9}, {3.4697157107808*^9, 3.469715745834*^9}, {3.4697157789372*^9, 3.4697160574128*^9}, { 3.469718813760184*^9, 3.4697188206865835`*^9}, {3.4697188540559835`*^9, 3.469718876863184*^9}, {3.469718919232784*^9, 3.469718949871184*^9}, { 3.4697189865623837`*^9, 3.4697191750103836`*^9}, {3.469719214400384*^9, 3.469719408807584*^9}, {3.4697194542659836`*^9, 3.4697195478815837`*^9}, { 3.469719585290384*^9, 3.469719614524784*^9}, {3.469719648704384*^9, 3.4697196558803835`*^9}, {3.4697196868931837`*^9, 3.4697196998099837`*^9}, {3.469719856433984*^9, 3.4697198616599836`*^9}, 3.4697200565039835`*^9, {3.4697200931951838`*^9, 3.469720174985984*^9}, { 3.4697202076835837`*^9, 3.469720211333984*^9}, {3.52637287878638*^9, 3.526372879909582*^9}}], Cell[CellGroupData[{ Cell["Sol", "Subsubsection", CellChangeTimes->{{3.4954666761469007`*^9, 3.495466677816104*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.466151159341*^9, 3.46615116226*^9}, 3.466151198587*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NMinimize", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"1.3", "*", RowBox[{ RowBox[{"(", RowBox[{"32", " ", "-", " ", "Ti"}], ")"}], "^", "2"}]}], " ", "+", " ", RowBox[{"3", "*", RowBox[{ RowBox[{"(", RowBox[{"0.3", " ", "-", " ", "hi"}], ")"}], "^", "2"}]}]}], ",", RowBox[{"20", "\[LessEqual]", " ", "Ti", "\[LessEqual]", "23"}], ",", " ", RowBox[{"0.5", "\[LessEqual]", " ", "hi", "\[LessEqual]", "0.6"}]}], "}"}], ",", RowBox[{"{", RowBox[{"Ti", ",", " ", "hi"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.465364518840598*^9, 3.465364523302198*^9}, { 3.4653646889741983`*^9, 3.4653647103929977`*^9}, {3.465364780265398*^9, 3.465364780483798*^9}, {3.465365073389398*^9, 3.465365090050198*^9}, { 3.465365184586198*^9, 3.465365185646998*^9}, {3.469719808994384*^9, 3.4697198474639835`*^9}, {3.469719949893584*^9, 3.4697199568043838`*^9}, { 3.469719989392784*^9, 3.469720004103584*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"105.42`", ",", RowBox[{"{", RowBox[{ RowBox[{"Ti", "\[Rule]", "23.`"}], ",", RowBox[{"hi", "\[Rule]", "0.5`"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.4955134583479795`*^9, 3.4955134965831666`*^9}, 3.497934560930927*^9, 3.497940403369195*^9, 3.4979410217477474`*^9, 3.5330136364006205`*^9, 3.533013918119916*^9, 3.5330139647673993`*^9, 3.533014985224715*^9}] }, Open ]], Cell["\<\ Los siguientes casos propuestos, y resueltos parcialmente, pueden ser \ requeridos por el profesor para utilizarlos como un dato m\[AAcute]s al \ evaluar la nota del alumno.\ \>", "Text", CellChangeTimes->{{3.5263730916643543`*^9, 3.5263731733304977`*^9}}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Inversi\[OAcute]n financiera ", "Subsection", CellChangeTimes->{{3.4950006345500026`*^9, 3.495000648804818*^9}, { 3.4950007202699056`*^9, 3.4950007786802464`*^9}, {3.4950009044144382`*^9, 3.495000937374323*^9}, {3.495001121995883*^9, 3.495001125360075*^9}, { 3.533054331988167*^9, 3.5330543386337786`*^9}}], Cell["\<\ Un grupo asesor financiero tiene un monto de 10 000 000 \[Euro] para ser \ colocado en varios productos financieros. El cuadro siguiente describe las 5 \ categor\[IAcute]as, con su respectivo beneficio y riesgo asociado.\ \>", "Text", CellChangeTimes->{{3.4950006345500026`*^9, 3.495000648804818*^9}, { 3.4950007202699056`*^9, 3.4950007786802464`*^9}, {3.4950009044144382`*^9, 3.495000937374323*^9}, {3.495001121995883*^9, 3.495001125360075*^9}, { 3.4950834393327937`*^9, 3.495083452171528*^9}, {3.4950835661880493`*^9, 3.495083572716423*^9}}], Cell[BoxData[GridBox[{ { RowBox[{"Producto", " ", RowBox[{"(", "i", ")"}]}], RowBox[{ RowBox[{"Beneficio", RowBox[{"(", "%", ")"}]}], ",", SubscriptBox["b", "i"], " "}], RowBox[{"Riesgo", " ", RowBox[{"(", "%", ")"}], SubscriptBox["r", "i"]}]}, { RowBox[{"1", "ra", " ", "Hipoteca"}], "9", "3"}, { RowBox[{"2", "nda", " ", "Hipoteca"}], "12", "6"}, { RowBox[{"Prestamos", " ", "personales"}], "15", "8"}, { RowBox[{"Prestamos", " ", "comerciales"}], "8", "2"}, { RowBox[{"Certificados", "/", "Bonos"}], "6", "1"} }, GridBoxDividers->{ "Columns" -> {{True}}, "ColumnsIndexed" -> {}, "Rows" -> {{True}}, "RowsIndexed" -> {}}]], "Text", CellChangeTimes->{{3.4950829911731606`*^9, 3.495083141611765*^9}, { 3.495083256677346*^9, 3.4950832972096643`*^9}, {3.495083427136096*^9, 3.4950834328844247`*^9}, {3.495083519744393*^9, 3.4950835804058623`*^9}, { 3.4988359997267165`*^9, 3.498836002867442*^9}}], Cell["\<\ El capital no invertido en alguna de estas categor\[IAcute]as, es colocado en \ deuda publica (sin riesgo) y con inter\[EAcute]s del 3%. El objetivo para el \ grupo asesor es asignar el dinero a cada una de las categor\[IAcute]as para \ cumplir con las metas siguientes : (a) Maximizar el beneficio por \[Euro] invertido (b) Que el riesgo promedio no supere el 5% ( sobre el dinero invertido) (c) Invertir al menos 20% en pr\[EAcute]stamos comerciales (d) El total invertido en 2nda Hipotecas y Prestamos Personales no podr\ \[AAcute] ser mayor que el invertido en 1era Hipotecas.\ \>", "Text", CellChangeTimes->{{3.4950006345500026`*^9, 3.495000648804818*^9}, { 3.4950007202699056`*^9, 3.4950007786802464`*^9}, {3.4950009762465467`*^9, 3.495001060524367*^9}, {3.495081114383814*^9, 3.495081149393817*^9}, { 3.4950831662041717`*^9, 3.4950832490899124`*^9}, {3.4950833236421766`*^9, 3.495083355140978*^9}}], Cell[CellGroupData[{ Cell["Sol", "Subsubsection", CellChangeTimes->{{3.4950827488863025`*^9, 3.495082750821413*^9}}], Cell[TextData[{ "Variables:", Cell[BoxData[ FormBox[ SubscriptBox["x", "i"], TraditionalForm]]], " es el porcentaje de la cantidad disponible que es invertida en el producto \ ", StyleBox["i. 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La parte invertida en deuda publica podemos considerarla un nuevo \ producto con beneficio, en %, ", Cell[BoxData[ FormBox[ SubscriptBox["b", "6"], TraditionalForm]]], "=3 y riesgo ", Cell[BoxData[ FormBox[ SubscriptBox["r", "6"], TraditionalForm]]], " =0 . 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En cada periodo se puede invertir en una o m\ \[AAcute]s de las opciones siguientes : - Opci\[OAcute]n A en una Caja de Ahorro con un interes anual de del 5%. - Opci\[OAcute]n Y inversi\[OAcute]n mantenida (maturity) por 2 a\[NTilde]os, \ con un retorno total de 12% al final del perido si compramos ahora o 11% m\ \[AAcute]s adelante. - Opci\[OAcute]n Z, inversi\[OAcute]n mantenida 3 a\[NTilde]os, y un retorno \ total de 18% al final del perido. - Opci\[OAcute]n W inversi\[OAcute]n mantenida 4 a\[NTilde]os, y un retorno \ total de 24% al final del perido. Se pueden hacer movimientos (dep\[OAcute]sitos /retiros) en la Caja de Ahorro \ en cualquier momento. Se pueden comprar Opciones Y todos los a\[NTilde]os, salvo en el a\[NTilde]o \ 3 . Se pueden comprar Opciones Z todos los a\[NTilde]os, despu\[EAcute]s del 1er \ a\[NTilde]o . La opci\[OAcute]n W, disponible en este momento, es una oportunidad \ \[UAcute]nica. (Con el objetivo de simplificar, se asumir\[AAcute] que cada opci\[OAcute]n \ puede se comprada en cualquier \[OpenCurlyDoubleQuote]denominaci\[OAcute]n\ \[CloseCurlyDoubleQuote]).\ \>", "Text", CellChangeTimes->{{3.4949980413746815`*^9, 3.49499805078922*^9}, { 3.494999399933387*^9, 3.494999434926388*^9}, {3.4949994683853016`*^9, 3.494999473034568*^9}, {3.494999936502076*^9, 3.4949999445895395`*^9}, { 3.4949999890070796`*^9, 3.495000210990776*^9}, {3.495000255252308*^9, 3.4950002686700754`*^9}, {3.495000309544413*^9, 3.4950003304346085`*^9}, { 3.495001065980679*^9, 3.49500109064809*^9}, {3.4950011437631283`*^9, 3.495001176179982*^9}, {3.4950014873437796`*^9, 3.4950015119271855`*^9}, { 3.49500156864743*^9, 3.4950015692244625`*^9}}], Cell["Graficamente el problema puede representarse como sigue :", "Text", CellChangeTimes->{{3.4950003814965286`*^9, 3.495000395728343*^9}}], Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 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La demanda diaria de cada uno \ de estos ciudades se muestra en la fila \"Demanda\" (cantidad m\[AAcute]xima \ que puede llegar a vender) es funci\[OAcute]n del tiempo t que va desde que \ dispone el pescado en la lonja hasta que lo entrega en la ciudad \ correspondiente (obs\[EAcute]rvese que la demanda va disminuyendo con el \ tiempo). Debe entregarlo entre 2 y 4 horas. Por otro lado tiene unos costes \ de trasporte que depende del tiempo de entrega dados por la funci\[OAcute]n \ indicada en la tabla (el coste sera mayor mientras m\[AAcute]s r\[AAcute]pido \ se haga la entrega) \[DownQuestion]C\[AAcute]lcula para que valor de t:{t1, \ t2, t3}, que representan el tiempo desde la lonja a C1, C2 y C3, que maximiza \ el beneficio?\[DownQuestion]Cual es el beneficio m\[AAcute]ximo esperado? \ \>", "Text", CellChangeTimes->{{3.4653876161546*^9, 3.4653877695025997`*^9}, { 3.4653878006714*^9, 3.4653878023406*^9}, {3.4653878398274*^9, 3.4653879191689997`*^9}, 3.465387983675*^9, {3.4653880698806*^9, 3.4653881776454*^9}, {3.4653882477986*^9, 3.4653883210406*^9}, { 3.4653884273078003`*^9, 3.4653884367925997`*^9}, {3.4653886325257998`*^9, 3.4653886531177998`*^9}, {3.465449932819189*^9, 3.465450207847189*^9}, { 3.465450315362389*^9, 3.4654503240203896`*^9}, {3.465450585351589*^9, 3.465450639920389*^9}, {3.465450675784789*^9, 3.465450687921589*^9}, { 3.4654508812055893`*^9, 3.4654509631367893`*^9}, {3.465452868505189*^9, 3.465452900609989*^9}, {3.4654529559899893`*^9, 3.4654529684387894`*^9}, { 3.4654530796979895`*^9, 3.4654531081367893`*^9}, {3.4654531515203896`*^9, 3.4654531573391895`*^9}, {3.4654533133391895`*^9, 3.4654533633839893`*^9}, {3.465453398515189*^9, 3.4654534061747894`*^9}, { 3.4654547236581893`*^9, 3.465454758165389*^9}, {3.4654553695605893`*^9, 3.4654554518973894`*^9}, {3.4654557641937895`*^9, 3.4654558356417894`*^9}, {3.465581760940788*^9, 3.465581815010388*^9}, { 3.465581847442788*^9, 3.4655818695947876`*^9}, {3.465581977078788*^9, 3.465582152235588*^9}, 3.465582187741188*^9, {3.4655832555455875`*^9, 3.4655832872447877`*^9}, {3.466154884524*^9, 3.466154885415*^9}, { 3.466154922399*^9, 3.4661550150699997`*^9}, {3.466178224139627*^9, 3.466178226152027*^9}, {3.466178788703627*^9, 3.4661788668284273`*^9}, { 3.4669492788449664`*^9, 3.466949297190566*^9}}], Cell[BoxData[ RowBox[{GridBox[{ { StyleBox[ RowBox[{"Ciudad", "->"}], FontSize->9], StyleBox[ RowBox[{"C1", " "}], FontSize->9], StyleBox[ RowBox[{"C2", " "}], FontSize->9], StyleBox[ RowBox[{"C3", " "}], FontSize->9]}, { StyleBox[ RowBox[{"Demanda", " ", RowBox[{"(", "ton", ")"}], " "}], FontSize->9], StyleBox[ RowBox[{"80", "-", RowBox[{"20", SuperscriptBox[ RowBox[{"(", RowBox[{"t1", "-", "2"}], ")"}], "1.2"]}]}], FontSize->9], StyleBox[ RowBox[{"180", "-", RowBox[{"30", SuperscriptBox[ RowBox[{"(", RowBox[{"t2", "-", "2"}], ")"}], "1.4"]}]}], FontSize->9], StyleBox[ RowBox[{"220", "-", RowBox[{"40", SuperscriptBox[ RowBox[{"(", RowBox[{"t3", "-", "2"}], ")"}], "1.6"]}]}], FontSize->9]}, { StyleBox[ RowBox[{ RowBox[{"Beneficio", ",", " ", RowBox[{"en", " ", "\[Euro]"}], ",", " ", "por"}], "\[IndentingNewLine]", RowBox[{"tonelada", RowBox[{"(", RowBox[{"sin", " ", "trasporte"}], ")"}]}]}], FontSize->9], StyleBox["3000", FontSize->9], StyleBox["3500", FontSize->9], StyleBox["4000", FontSize->9]}, { StyleBox[ RowBox[{"Coste", " ", "total", " ", "de", " ", "trasporte", " ", RowBox[{"(", RowBox[{"en", " ", "\[Euro]"}], ")"}]}], FontSize->9], StyleBox[ RowBox[{"800", SuperscriptBox[ RowBox[{"(", RowBox[{"t1", "-", "2"}], ")"}], RowBox[{"-", "5"}]]}], FontSize->9], StyleBox[ RowBox[{"2000", SuperscriptBox[ RowBox[{"(", RowBox[{"t2", "-", "2"}], ")"}], RowBox[{"-", "5"}]]}], FontSize->9], StyleBox[ RowBox[{"3000", SuperscriptBox[ RowBox[{"(", RowBox[{"t3", "-", "2"}], ")"}], RowBox[{"-", "5"}]]}], FontSize->9]} }, GridBoxDividers->{ "Columns" -> {{True}}, "ColumnsIndexed" -> {}, "Rows" -> {{True}}, "RowsIndexed" -> {}}], "\[NoBreak]"}]], "Text", CellChangeTimes->{{3.4654552475061893`*^9, 3.4654553296089892`*^9}, { 3.465455485312589*^9, 3.465455491022189*^9}, {3.465455529444989*^9, 3.4654555406925893`*^9}, {3.465582272963988*^9, 3.465582279640788*^9}, { 3.4655826973151875`*^9, 3.465582818277588*^9}}], Cell[CellGroupData[{ Cell["Sol", "Subsubsection", CellChangeTimes->{{3.465587235745988*^9, 3.4655872363543878`*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.466151159341*^9, 3.46615116226*^9}, 3.466151198587*^9}], Cell[TextData[{ "El beneficio neto vendr\[AAcute] dado por el beneficio menos los costes de \ trasporte. Si llamamos ", Cell[BoxData[ FormBox[ SubscriptBox["BN", "i"], TraditionalForm]]], " al beneficio neto, ", Cell[BoxData[ FormBox[ SubscriptBox["D", "i"], TraditionalForm]]], " a la demanda, en toneladas, ", Cell[BoxData[ FormBox[ SubscriptBox["B", "i"], TraditionalForm]]], " al beneficio por tonelada y ", Cell[BoxData[ FormBox[ SubscriptBox["CT", "i"], TraditionalForm]]], " al coste de transporte, siendo i: {C1, C2, C3}. Se trata por tanto de \ minimizar ", Cell[BoxData[ FormBox[ RowBox[{"\[Sum]", " ", SubscriptBox["BN", "i"]}], TraditionalForm]]], " =\[Sum](", Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["B", "i"], " ", SubscriptBox["D", "i"], RowBox[{"(", "t", ")"}]}], "-"}]]], " ", Cell[BoxData[ RowBox[{ SubscriptBox["CT", "i"], RowBox[{"(", "t", ")"}]}]], CellChangeTimes->{3.466949379995366*^9, 3.4669494102281666`*^9}], ")" }], "Text", CellChangeTimes->{{3.465455593248989*^9, 3.4654556225925894`*^9}, { 3.4654556546817894`*^9, 3.465455750980589*^9}, {3.465581717697588*^9, 3.465581722892388*^9}, 3.4655827069403877`*^9, {3.4655828564663877`*^9, 3.465583205594388*^9}, 3.465583246762788*^9, {3.4661550483269997`*^9, 3.466155057888*^9}, {3.4661782736072273`*^9, 3.4661783448680267`*^9}, { 3.466178388329627*^9, 3.4661783911376266`*^9}, {3.4669492939613667`*^9, 3.4669493443025665`*^9}, {3.4669494313671665`*^9, 3.4669494703671665`*^9}, {3.4979331602858143`*^9, 3.4979332474397993`*^9}, {3.4979332947485056`*^9, 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cal (en \ toneladas), mineral (en toneladas, t) y mano de obra (en horas). Los \ costes son: 1 t de cal 23\[LineSeparator] euros, 1 t de mineral 20 euros y 1 \ hora de mano de obra 23 euros. A esos precios sl mes se dispone como mucho \ 3000 t de cal y 4000 t de mineral al mes y 2000 horas. \ \>", "Text", CellChangeTimes->{{3.5303736170421276`*^9, 3.5303736268389444`*^9}, { 3.530373666538213*^9, 3.5303737516051626`*^9}, {3.530373815643275*^9, 3.5303738284040976`*^9}, {3.530373882426992*^9, 3.5303739150466495`*^9}, { 3.5303739460751038`*^9, 3.5303740122636194`*^9}, {3.5303740657873135`*^9, 3.5303740903261566`*^9}, {3.5303741836611204`*^9, 3.5303742008523507`*^9}, {3.530374233596808*^9, 3.5303744924300623`*^9}, { 3.530374621465488*^9, 3.530374801755005*^9}, {3.5303748652847157`*^9, 3.5303748973427715`*^9}, {3.5303749785721145`*^9, 3.5303749797733164`*^9}, {3.530375010832971*^9, 3.5303750280530005`*^9}, { 3.5305351494048977`*^9, 3.5305353034863687`*^9}, {3.5305353950273294`*^9, 3.5305355294059653`*^9}, {3.5305356055496993`*^9, 3.530535649775777*^9}, { 3.530536262466853*^9, 3.530536264151656*^9}, {3.5305363407477903`*^9, 3.530536427624343*^9}, {3.530536587961425*^9, 3.530536608740661*^9}, 3.5305367127148438`*^9, {3.5305368656575127`*^9, 3.5305368668275146`*^9}, { 3.530536911568393*^9, 3.530536912473195*^9}}], Cell["\<\ a) El precio de venta del cobre es 36 euros/kg. \[DownQuestion]Qu\[EAcute] \ valores deben tener u, v y w para maximizar el beneficio? \[DownQuestion]C\ \[UAcute]al ser\[AAcute] es beneficio?\ \>", "Text", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmViYGAQAWIQLROpaLfc95XjohtR9iCayzLQBUTXxX72A9Fbp34J A9Fx73oiQPTrlb7xIFq9LjEZRDukO6WD6En5Hfkgeo2hZRmIfpDvVQmin6l/ bQHRpW7S7SCaV9K4G0QLHe5dDaLdn67bDaIZNC0ugegUGc1bIPrPd617IPqw X9ALEM20vwJMb14e9QZE621qfAei33xZI/sNSIdOeG8Aoh/dT7AB0av/6viC aOXpG8NAtNZlixgQXfbh3AoQHXeLZSWIfvZ2+iYQPe9X8V4Q/e9w3UUQXb9Z /AqIlrGoeAiiz4VdewKiWRS4pb+D9K+JlQXRANoeq9c= "]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NMaximize", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"36", SqrtBox["u"], " ", SuperscriptBox["v", RowBox[{"1", "/", "3"}]], " ", SuperscriptBox["w", RowBox[{"1", "/", "4"}]]}], " ", "-", RowBox[{"25", "u"}], "-", RowBox[{"20", "v"}], "-", RowBox[{"23", "w"}]}], ",", RowBox[{"0", "<", "u", "<", "3000"}], ",", RowBox[{"0", "<", "v", "<", "4000"}], ",", RowBox[{"0", "<", "w", "<", "2000"}]}], "}"}], ",", RowBox[{"{", RowBox[{"u", ",", "v", ",", "w"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.530372019603326*^9, 3.530372181261608*^9}, { 3.530372288574197*^9, 3.5303722964366107`*^9}, {3.5303723730639453`*^9, 3.5303726799284835`*^9}, {3.530373052844738*^9, 3.5303731021720247`*^9}, { 3.530373367042489*^9, 3.530373398008543*^9}, {3.5303734523746386`*^9, 3.530373535631985*^9}, {3.5303744193619347`*^9, 3.5303744245567436`*^9}, { 3.5303748201942368`*^9, 3.530374855877899*^9}, {3.5305353090087786`*^9, 3.530535342564437*^9}, {3.5305356641590023`*^9, 3.5305360584496946`*^9}, { 3.530536094626158*^9, 3.5305362416408167`*^9}, {3.5305362917325044`*^9, 3.5305363186113515`*^9}, {3.530536356691018*^9, 3.5305363657858343`*^9}, { 3.5305364591519985`*^9, 3.530536490352053*^9}, {3.530536531005725*^9, 3.530536573094599*^9}, {3.5305366955704136`*^9, 3.5305366966468153`*^9}, { 3.530536855205494*^9, 3.530536855564295*^9}, {3.530537168984445*^9, 3.530537184116472*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"9320.895725654264`", ",", RowBox[{"{", RowBox[{ RowBox[{"u", "\[Rule]", "2999.9999999991646`"}], ",", RowBox[{"v", "\[Rule]", "3258.0223931349406`"}], ",", RowBox[{"w", "\[Rule]", "1999.9999999945226`"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.5303720832334366`*^9, 3.530372119815501*^9}, { 3.5303721556931634`*^9, 3.5303721823692102`*^9}, 3.5303722974038124`*^9, 3.530372393577981*^9, {3.530372428724843*^9, 3.5303724341068525`*^9}, 3.530372470014515*^9, 3.5303725307922215`*^9, {3.5303725627566776`*^9, 3.5303726811452856`*^9}, 3.530373098178417*^9, {3.5303733952161384`*^9, 3.530373400270547*^9}, {3.5303734580998487`*^9, 3.5303735375195885`*^9}, 3.5303744272227483`*^9, 3.530374857562702*^9, 3.5305353518932533`*^9, { 3.5305357033462706`*^9, 3.5305357179166965`*^9}, {3.5305357502867537`*^9, 3.530535827397689*^9}, {3.530535859252945*^9, 3.5305359121214375`*^9}, { 3.5305359429158916`*^9, 3.5305360594012966`*^9}, 3.530536102566572*^9, { 3.5305361335482264`*^9, 3.5305362427484183`*^9}, {3.5305362930741067`*^9, 3.530536319469353*^9}, 3.53053635764262*^9, {3.5305364484659796`*^9, 3.530536491334855*^9}, {3.530536532565727*^9, 3.5305365739526*^9}, 3.5305367463797026`*^9, 3.530536856391096*^9, {3.5305371715584497`*^9, 3.5305371851772738`*^9}}] }, Open ]], Cell["\<\ b) El precio del cobre baja a 34 euros kg \[DownQuestion]Obtengo beneficio? 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