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\raisebox{-.5\height}{\includegraphics[width=5cm]{immagini/ProblemaDuale1.jpg}} \end{center} -Osservando l'ultima immagine per chiarezza col risultato finale che si riconduce al sistema iniziale. +Osservando l'ultima immagine per chiarezza col risultato finale che si riconduce al sistema iniziale.\\ +Se volessi risolvere il sistema invece, partendo dal sistema già orientato nel verso giusto e il suo problema duale:\\ + +\begin{align*} +min: &2x_1 + 4x_2 - 3x_3 & & & max: &5u_1 - 10u_2 \\ +&x_1 - 3x_2 - x_3 \geqq -5 & & & &u_1 - u_2 \leqq 2\\ +&-2x_1 - 2x_2 + x_3 \geqq -10 & & & &-3u_1 - 2u_2 \leqq 4\\ +&x_1,x_2,x_3 \geqq 0 & & & &-u_1 + u_2 \leqq -3\\ +& & & & &u_1,u_2 \geqq 0\\ +\end{align*} + +\begin{align*} +\left\{ \begin{array}{rcl} (x_1 -3x_2 - x_3 + 5)u_1 = 0 \\ (-2x_1 -2x_2 - x_3 + 10)u_2 = 0 \\ (u_1 -2u_2 - 2 )x_1 = 0 \\ (-3u_1 -2u_2 - 4 )x_2 = 0 \\ (-u_1 +u_2 + 3 )x_3 = 0 \end{array}\right. & &\Rightarrow & &\left\{ \begin{array}{rcl} (0)u_1 = 0 \\ (0)u_2 = 0 \\ u_1 -2u_2 = 2 \\ (-3u_1 -2u_2 - 2 )0 = 0 \\ (-u_1 +u_2) = -3 \end{array}\right.\\ +\end{align*} +$u = (4,1) z_D= -30$ + +\begin{itemize} +\item Moltiplico ogni vincolo del primale per la sua variabile corrispondente del +duale e lo pongo uguale a zero, prima porto a sinistra il termine noto.\\ +\item Moltiplico ogni vincolo del duale per la sua variabili corrispondente del +primale e lo pongo uguale a zero, prima porto a sinistra il termine noto. +\item Metto tutto a sistema e semplifico, ad esempio sappiamo i valori delle +variabili della funzione obiettivo del duale, inoltre i vincoli che ci portano alla +soluzione ottima saranno uguale a zero e i restanti diversi da \item Ottengo un sistema che posso risolvere, e dovrebbero essere presenti solo +variabili del problema primale, risolvo il sistema e ottengo i valori delle +variabili della funzione obiettivo del problema primale +\end{itemize} \subsection{Rappresentazione Grafica} Anche tu sei a digiuno di grafici ed hai un incontrollato appetito di disegnarne altri ?\\ diff --git a/SimpleDocument.toc b/SimpleDocument.toc index 263bb1c..c95b2b9 100644 --- a/SimpleDocument.toc +++ b/SimpleDocument.toc @@ -19,46 +19,46 @@ \contentsline {subsubsection}{\numberline {3.2.2}Simplesso Duale}{16}{subsubsection.3.2.2}% \contentsline {subsubsection}{\numberline {3.2.3}2Fasi}{17}{subsubsection.3.2.3}% \contentsline {subsubsection}{\numberline {3.2.4}Duale del problema}{19}{subsubsection.3.2.4}% -\contentsline {subsection}{\numberline {3.3}Rappresentazione Grafica}{21}{subsection.3.3}% -\contentsline {subsection}{\numberline {3.4}ILP}{22}{subsection.3.4}% -\contentsline {subsubsection}{\numberline {3.4.1}Tagli di Gomory}{22}{subsubsection.3.4.1}% +\contentsline {subsection}{\numberline {3.3}Rappresentazione Grafica}{22}{subsection.3.3}% +\contentsline {subsection}{\numberline {3.4}ILP}{23}{subsection.3.4}% +\contentsline {subsubsection}{\numberline {3.4.1}Tagli di Gomory}{23}{subsubsection.3.4.1}% \contentsline {subsection}{\numberline {3.5}Esercizi Particolari}{25}{subsection.3.5}% \contentsline {subsubsection}{\numberline {3.5.1}Simplesso con variabili free}{25}{subsubsection.3.5.1}% -\contentsline {subsubsection}{\numberline {3.5.2}Problema PLC con simplesso e gomory in salsa teriyaki}{27}{subsubsection.3.5.2}% -\contentsline {subsubsection}{\numberline {3.5.3}Senstivity Analysis}{29}{subsubsection.3.5.3}% -\contentsline {section}{\numberline {4}Grafi}{31}{section.4}% -\contentsline {subsection}{\numberline {4.1}GT}{31}{subsection.4.1}% -\contentsline {subsubsection}{\numberline {4.1.1}Dijkstra}{31}{subsubsection.4.1.1}% -\contentsline {subsubsection}{\numberline {4.1.2}Shortest Path Tree}{33}{subsubsection.4.1.2}% -\contentsline {subsection}{\numberline {4.2}SST}{34}{subsection.4.2}% -\contentsline {subsubsection}{\numberline {4.2.1}Cenni di Base}{34}{subsubsection.4.2.1}% -\contentsline {subsubsection}{\numberline {4.2.2}SST Prim's}{35}{subsubsection.4.2.2}% -\contentsline {subsubsection}{\numberline {4.2.3}SST Da Matrice trovare la soluzione ottimale}{36}{subsubsection.4.2.3}% -\contentsline {subsection}{\numberline {4.3}Max Flow}{37}{subsection.4.3}% -\contentsline {subsubsection}{\numberline {4.3.1}Cenni di Base}{37}{subsubsection.4.3.1}% -\contentsline {subsubsection}{\numberline {4.3.2}Flow Network}{38}{subsubsection.4.3.2}% -\contentsline {subsection}{\numberline {4.4}DP}{40}{subsection.4.4}% -\contentsline {subsubsection}{\numberline {4.4.1}DP Knapsack 0-1 Dynamic Programming}{40}{subsubsection.4.4.1}% -\contentsline {subsubsection}{\numberline {4.4.2}DP: SSP Bellman's-Ford}{42}{subsubsection.4.4.2}% -\contentsline {subsection}{\numberline {4.5}ILP}{44}{subsection.4.5}% -\contentsline {subsubsection}{\numberline {4.5.1}ILP standard B\&B}{44}{subsubsection.4.5.1}% -\contentsline {subsubsection}{\numberline {4.5.2}ILP esercizio standard B\&B}{48}{subsubsection.4.5.2}% -\contentsline {subsubsection}{\numberline {4.5.3}ILP esercizio B\&B 0-1}{50}{subsubsection.4.5.3}% -\contentsline {section}{\numberline {5}GPLK}{52}{section.5}% -\contentsline {subsection}{\numberline {5.1}Dal modello al codice}{52}{subsection.5.1}% -\contentsline {subsubsection}{\numberline {5.1.1}Modello Easy}{52}{subsubsection.5.1.1}% -\contentsline {subsubsection}{\numberline {5.1.2}Caso Particolare 1 Graffa}{53}{subsubsection.5.1.2}% -\contentsline {subsubsection}{\numberline {5.1.3}Caso Particolare 2 Sommatoria doppio insieme}{55}{subsubsection.5.1.3}% -\contentsline {subsection}{\numberline {5.2}Dal codice al modello}{56}{subsection.5.2}% -\contentsline {subsubsection}{\numberline {5.2.1}Normale}{56}{subsubsection.5.2.1}% -\contentsline {section}{\numberline {6}Domande varie di teoria}{57}{section.6}% -\contentsline {subsection}{\numberline {6.1}PLC degenerate}{57}{subsection.6.1}% -\contentsline {subsection}{\numberline {6.2}PLC sensitivty}{58}{subsection.6.2}% -\contentsline {subsection}{\numberline {6.3}B\&B}{58}{subsection.6.3}% -\contentsline {subsection}{\numberline {6.4}PLC minimization}{58}{subsection.6.4}% -\contentsline {subsection}{\numberline {6.5}cutting Plane}{58}{subsection.6.5}% -\contentsline {subsection}{\numberline {6.6}PLC dual}{59}{subsection.6.6}% -\contentsline {subsection}{\numberline {6.7}Dijkstra}{59}{subsection.6.7}% -\contentsline {subsection}{\numberline {6.8}B\&B knapsack}{59}{subsection.6.8}% -\contentsline {subsection}{\numberline {6.9}Branch \& cut}{59}{subsection.6.9}% -\contentsline {subsection}{\numberline {6.10}PLC}{60}{subsection.6.10}% +\contentsline {subsubsection}{\numberline {3.5.2}Problema PLC con simplesso e gomory in salsa teriyaki}{28}{subsubsection.3.5.2}% +\contentsline {subsubsection}{\numberline {3.5.3}Senstivity Analysis}{30}{subsubsection.3.5.3}% +\contentsline {section}{\numberline {4}Grafi}{32}{section.4}% +\contentsline {subsection}{\numberline {4.1}GT}{32}{subsection.4.1}% +\contentsline {subsubsection}{\numberline {4.1.1}Dijkstra}{32}{subsubsection.4.1.1}% +\contentsline {subsubsection}{\numberline {4.1.2}Shortest Path Tree}{34}{subsubsection.4.1.2}% +\contentsline {subsection}{\numberline {4.2}SST}{35}{subsection.4.2}% +\contentsline {subsubsection}{\numberline {4.2.1}Cenni di Base}{35}{subsubsection.4.2.1}% +\contentsline {subsubsection}{\numberline {4.2.2}SST Prim's}{36}{subsubsection.4.2.2}% +\contentsline {subsubsection}{\numberline {4.2.3}SST Da Matrice trovare la soluzione ottimale}{37}{subsubsection.4.2.3}% +\contentsline {subsection}{\numberline {4.3}Max Flow}{38}{subsection.4.3}% +\contentsline {subsubsection}{\numberline {4.3.1}Cenni di Base}{38}{subsubsection.4.3.1}% +\contentsline {subsubsection}{\numberline {4.3.2}Flow Network}{39}{subsubsection.4.3.2}% +\contentsline {subsection}{\numberline {4.4}DP}{41}{subsection.4.4}% +\contentsline {subsubsection}{\numberline {4.4.1}DP Knapsack 0-1 Dynamic Programming}{41}{subsubsection.4.4.1}% +\contentsline {subsubsection}{\numberline {4.4.2}DP: SSP Bellman's-Ford}{43}{subsubsection.4.4.2}% +\contentsline {subsection}{\numberline {4.5}ILP}{45}{subsection.4.5}% +\contentsline {subsubsection}{\numberline {4.5.1}ILP standard B\&B}{45}{subsubsection.4.5.1}% +\contentsline {subsubsection}{\numberline {4.5.2}ILP esercizio standard B\&B}{49}{subsubsection.4.5.2}% +\contentsline {subsubsection}{\numberline {4.5.3}ILP esercizio B\&B 0-1}{51}{subsubsection.4.5.3}% +\contentsline {section}{\numberline {5}GPLK}{53}{section.5}% +\contentsline {subsection}{\numberline {5.1}Dal modello al codice}{53}{subsection.5.1}% +\contentsline {subsubsection}{\numberline {5.1.1}Modello Easy}{53}{subsubsection.5.1.1}% +\contentsline {subsubsection}{\numberline {5.1.2}Caso Particolare 1 Graffa}{54}{subsubsection.5.1.2}% +\contentsline {subsubsection}{\numberline {5.1.3}Caso Particolare 2 Sommatoria doppio insieme}{56}{subsubsection.5.1.3}% +\contentsline {subsection}{\numberline {5.2}Dal codice al modello}{57}{subsection.5.2}% +\contentsline {subsubsection}{\numberline {5.2.1}Normale}{57}{subsubsection.5.2.1}% +\contentsline {section}{\numberline {6}Domande varie di teoria}{58}{section.6}% +\contentsline {subsection}{\numberline {6.1}PLC degenerate}{58}{subsection.6.1}% +\contentsline {subsection}{\numberline {6.2}PLC sensitivty}{59}{subsection.6.2}% +\contentsline {subsection}{\numberline {6.3}B\&B}{59}{subsection.6.3}% +\contentsline {subsection}{\numberline {6.4}PLC minimization}{59}{subsection.6.4}% +\contentsline {subsection}{\numberline {6.5}cutting Plane}{59}{subsection.6.5}% +\contentsline {subsection}{\numberline {6.6}PLC dual}{60}{subsection.6.6}% +\contentsline {subsection}{\numberline {6.7}Dijkstra}{60}{subsection.6.7}% +\contentsline {subsection}{\numberline {6.8}B\&B knapsack}{60}{subsection.6.8}% +\contentsline {subsection}{\numberline {6.9}Branch \& cut}{60}{subsection.6.9}% +\contentsline {subsection}{\numberline {6.10}PLC}{61}{subsection.6.10}%