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[2][-]{subsection.6.10}{PLC}{section.6}% 64 diff --git a/SimpleDocument.pdf b/SimpleDocument.pdf index 3cc6f95..1af510a 100644 Binary files a/SimpleDocument.pdf and b/SimpleDocument.pdf differ diff --git a/SimpleDocument.synctex.gz b/SimpleDocument.synctex.gz index e93b077..3cf15fd 100644 Binary files a/SimpleDocument.synctex.gz and b/SimpleDocument.synctex.gz differ diff --git a/SimpleDocument.tex b/SimpleDocument.tex index aaa3fd8..1058fc4 100644 --- a/SimpleDocument.tex +++ b/SimpleDocument.tex @@ -3,21 +3,23 @@ %Headers \usepackage[dvips]{graphicx} %package that does pdfs \usepackage{color} %this needs to be here also -\usepackage{amssymb} +\usepackage{amssymb} %librerie simboli matematica \usepackage{amsmath} -\usepackage{xcolor,listings} +\usepackage{xcolor,listings} %per il codice \usepackage{textcomp} \lstset{upquote=true} -\usepackage{imakeidx} +\usepackage{hyperref} %per gli hyperlink + +\usepackage{imakeidx} %per l'indice \makeindex \title{Gran Compendio OLI} -\author{Meschio} -\date{2021\\V1.0} +\author{\href{https://github.com/meschio94/Gran-Compendio-OLI}{Meschio}} +\date{2021\\V1.0a} \begin{document} \maketitle @@ -1358,6 +1360,8 @@ Soluzione:\\ Given a primal PLC problem in minimization form and its dual: (a) if the primal has finite solution, then the dual has finite solution; (b) a feasible dual solution may have a value smaller than that of the optimal primal solution; (c) if the primal is unbounded, then the dual is unbounded; (d) any feasible solution of the primal has a value greater or equal to any feasible solution of the dual; (e) if the primal is empty, then the dual is empty. }% }% +Soluzione:\\ +(A,B,D) \subsection{Dijkstra} \noindent\fbox{% @@ -1365,6 +1369,8 @@ Given a primal PLC problem in minimization form and its dual: (a) if the primal The Dijkstra algorithm : (a) cannot be used in a graph with negative cost arcs; (b) cannot be used in a graph with negative circuits; (c) can be used to find the shortest path in a directed graph, from a source node to a sink node; (d) can be used to find the minimum cost spanning tree; (e) cannot be applied to acyclic graphs. }% }% +Soluzione:\\ +(A,B,C) \subsection{B\&B knapsack} \noindent\fbox{% @@ -1372,6 +1378,8 @@ The Dijkstra algorithm : (a) cannot be used in a graph with negative cost arcs; The branch-and-bound method for 0-1 knapsack problems: (a) uses a decision tree with an exponential number of levels; (b) uses a decision tree with a polynomial number of levels; (c) uses the lowest-first strategy to explore the decision tree; (d) computes an upper bound at each node of the decision tree; (e) uses a decision tree with an exponential number of nodes. }% }% +Soluzione:\\ +(B,E) \subsection{Branch \& cut} \noindent\fbox{% @@ -1379,12 +1387,14 @@ The branch-and-bound method for 0-1 knapsack problems: (a) uses a decision tree The branch-and-cut method: (a) is used to solve linear programming problems with continuous variables; (b) always uses the lowest-first strategy to explore the subproblems.; (c) is used to solve NP-complete (“difficult”) problems; (d) terminates in a polynomial number of iterations; (e) at each iteration uses the simplex algorithm to solve the current subproblem. }% }% - +Soluzione:\\ +(C,E) \subsection{PLC} \noindent\fbox{% \parbox{\textwidth}{% Given a PLC problem with $n$ variables and $m$ constraints, a basic matrix is: (a) a collection of $m$ constraints; (b) none of the above answers.; (c) a square $m$x$m$ matrix with value 1 on the main diagonal; (d) a collection of $n-m$ columns of the constraint matrix; (e) a square submatrix of the constraint matrix, that can be inverted. }% }% - +Soluzione:\\ +(B) \end{document} diff --git a/SimpleDocument.toc b/SimpleDocument.toc index f7c45bb..430348d 100644 --- a/SimpleDocument.toc +++ b/SimpleDocument.toc @@ -1,64 +1,64 @@ -\contentsline {section}{\numberline {1}Modelli}{3}{}% -\contentsline {subsection}{\numberline {1.1}Cenni di Base}{3}{}% -\contentsline {subsection}{\numberline {1.2}Esercizi}{5}{}% -\contentsline {subsubsection}{\numberline {1.2.1}Problema con Delta}{5}{}% -\contentsline {subsubsection}{\numberline {1.2.2}Problema con massima distanza}{6}{}% -\contentsline {subsubsection}{\numberline {1.2.3}Problema Variabile Triplo indice}{7}{}% -\contentsline {subsubsection}{\numberline {1.2.4}Problema di Trasporto}{9}{}% -\contentsline {subsubsection}{\numberline {1.2.5}Problema di Stoccaggio}{11}{}% -\contentsline {subsubsection}{\numberline {1.2.6}Problema di Stoccaggio 2}{12}{}% -\contentsline {section}{\numberline {2}Forme: Standard, Canonica, Generale}{13}{}% -\contentsline {subsection}{\numberline {2.1}Standard $\Rightarrow $ Canonica}{13}{}% -\contentsline {subsection}{\numberline {2.2}Generale $\Rightarrow $ Standard}{13}{}% -\contentsline {subsection}{\numberline {2.3}Esercizio Trasformazione Gran Fritto Misto}{14}{}% -\contentsline {section}{\numberline {3}Matrici : LP, ILP}{15}{}% -\contentsline {subsection}{\numberline {3.1}Fondamenti Concettuali}{15}{}% -\contentsline {subsubsection}{\numberline {3.1.1}Quale Simplesso?}{15}{}% -\contentsline {subsection}{\numberline {3.2}LP}{15}{}% -\contentsline {subsubsection}{\numberline {3.2.1}Simplesso}{15}{}% -\contentsline {subsubsection}{\numberline {3.2.2}Simplesso Duale}{16}{}% -\contentsline {subsubsection}{\numberline {3.2.3}2Fasi}{17}{}% -\contentsline {subsubsection}{\numberline {3.2.4}Duale del problema}{19}{}% -\contentsline {subsection}{\numberline {3.3}Rappresentazione Grafica}{20}{}% -\contentsline {subsection}{\numberline {3.4}ILP}{22}{}% -\contentsline {subsubsection}{\numberline {3.4.1}Tagli di Gomory}{22}{}% -\contentsline {subsection}{\numberline {3.5}Esercizi Particolari}{24}{}% -\contentsline {subsubsection}{\numberline {3.5.1}Simplesso con variabili free}{24}{}% -\contentsline {subsubsection}{\numberline {3.5.2}Problema PLC con simplesso e gomory in salsa teriyaki}{26}{}% -\contentsline {subsubsection}{\numberline {3.5.3}Senstivity Analysis}{28}{}% -\contentsline {section}{\numberline {4}Grafi}{30}{}% -\contentsline {subsection}{\numberline {4.1}GT}{30}{}% -\contentsline {subsubsection}{\numberline {4.1.1}Dijkstra}{30}{}% -\contentsline {subsubsection}{\numberline {4.1.2}Shortest Path Tree}{32}{}% -\contentsline {subsection}{\numberline {4.2}SST}{33}{}% -\contentsline {subsubsection}{\numberline {4.2.1}Cenni di Base}{33}{}% -\contentsline {subsubsection}{\numberline {4.2.2}SST Prim's}{34}{}% -\contentsline {subsubsection}{\numberline {4.2.3}SST Da Matrice trovare la soluzione ottimale}{35}{}% -\contentsline {subsection}{\numberline {4.3}Max Flow}{36}{}% -\contentsline {subsubsection}{\numberline {4.3.1}Cenni di Base}{36}{}% -\contentsline {subsubsection}{\numberline {4.3.2}Flow Network}{37}{}% -\contentsline {subsection}{\numberline {4.4}DP}{39}{}% -\contentsline {subsubsection}{\numberline {4.4.1}DP Knapsack 0-1 Dynamic Programming}{39}{}% -\contentsline {subsubsection}{\numberline {4.4.2}DP: SSP Bellman's-Ford}{41}{}% -\contentsline {subsection}{\numberline {4.5}ILP}{43}{}% -\contentsline {subsubsection}{\numberline {4.5.1}ILP standard B\&B}{43}{}% -\contentsline {subsubsection}{\numberline {4.5.2}ILP esercizio standard B\&B}{47}{}% -\contentsline {subsubsection}{\numberline {4.5.3}ILP esercizio B\&B 0-1}{49}{}% -\contentsline {section}{\numberline {5}GPLK}{51}{}% -\contentsline {subsection}{\numberline {5.1}Dal modello al codice}{51}{}% -\contentsline {subsubsection}{\numberline {5.1.1}Modello Easy}{51}{}% -\contentsline {subsubsection}{\numberline {5.1.2}Caso Particolare 1 Graffa}{52}{}% -\contentsline {subsubsection}{\numberline {5.1.3}Caso Particolare 2 Sommatoria doppio insieme}{54}{}% -\contentsline {subsection}{\numberline {5.2}Dal codice al modello}{55}{}% -\contentsline {subsubsection}{\numberline {5.2.1}Normale}{55}{}% -\contentsline {section}{\numberline {6}Domande varie di teoria}{56}{}% -\contentsline {subsection}{\numberline {6.1}PLC degenerate}{56}{}% -\contentsline {subsection}{\numberline {6.2}PLC sensitivty}{57}{}% -\contentsline {subsection}{\numberline {6.3}B\&B}{57}{}% -\contentsline {subsection}{\numberline {6.4}PLC minimization}{57}{}% -\contentsline {subsection}{\numberline {6.5}cutting Plane}{57}{}% -\contentsline {subsection}{\numberline {6.6}PLC dual}{58}{}% -\contentsline {subsection}{\numberline {6.7}Dijkstra}{58}{}% -\contentsline {subsection}{\numberline {6.8}B\&B knapsack}{58}{}% -\contentsline {subsection}{\numberline {6.9}Branch \& cut}{58}{}% -\contentsline {subsection}{\numberline {6.10}PLC}{58}{}% +\contentsline {section}{\numberline {1}Modelli}{1}{section.1}% +\contentsline {subsection}{\numberline {1.1}Cenni di Base}{1}{subsection.1.1}% +\contentsline {subsection}{\numberline {1.2}Esercizi}{4}{subsection.1.2}% +\contentsline {subsubsection}{\numberline {1.2.1}Problema con Delta}{4}{subsubsection.1.2.1}% +\contentsline {subsubsection}{\numberline {1.2.2}Problema con massima distanza}{5}{subsubsection.1.2.2}% +\contentsline {subsubsection}{\numberline {1.2.3}Problema Variabile Triplo indice}{6}{subsubsection.1.2.3}% +\contentsline {subsubsection}{\numberline {1.2.4}Problema di Trasporto}{8}{subsubsection.1.2.4}% +\contentsline {subsubsection}{\numberline {1.2.5}Problema di Stoccaggio}{10}{subsubsection.1.2.5}% +\contentsline {subsubsection}{\numberline {1.2.6}Problema di Stoccaggio 2}{11}{subsubsection.1.2.6}% +\contentsline {section}{\numberline {2}Forme: Standard, Canonica, Generale}{12}{section.2}% +\contentsline {subsection}{\numberline {2.1}Standard $\Rightarrow $ Canonica}{12}{subsection.2.1}% +\contentsline {subsection}{\numberline {2.2}Generale $\Rightarrow $ Standard}{12}{subsection.2.2}% +\contentsline {subsection}{\numberline {2.3}Esercizio Trasformazione Gran Fritto Misto}{13}{subsection.2.3}% +\contentsline {section}{\numberline {3}Matrici : LP, ILP}{14}{section.3}% +\contentsline {subsection}{\numberline {3.1}Fondamenti Concettuali}{14}{subsection.3.1}% +\contentsline {subsubsection}{\numberline {3.1.1}Quale Simplesso?}{14}{subsubsection.3.1.1}% +\contentsline {subsection}{\numberline {3.2}LP}{14}{subsection.3.2}% +\contentsline {subsubsection}{\numberline {3.2.1}Simplesso}{14}{subsubsection.3.2.1}% +\contentsline {subsubsection}{\numberline {3.2.2}Simplesso Duale}{15}{subsubsection.3.2.2}% +\contentsline {subsubsection}{\numberline {3.2.3}2Fasi}{16}{subsubsection.3.2.3}% +\contentsline {subsubsection}{\numberline {3.2.4}Duale del problema}{18}{subsubsection.3.2.4}% +\contentsline {subsection}{\numberline {3.3}Rappresentazione Grafica}{19}{subsection.3.3}% +\contentsline {subsection}{\numberline {3.4}ILP}{21}{subsection.3.4}% +\contentsline {subsubsection}{\numberline {3.4.1}Tagli di Gomory}{21}{subsubsection.3.4.1}% +\contentsline {subsection}{\numberline {3.5}Esercizi Particolari}{23}{subsection.3.5}% +\contentsline {subsubsection}{\numberline {3.5.1}Simplesso con variabili free}{23}{subsubsection.3.5.1}% +\contentsline {subsubsection}{\numberline {3.5.2}Problema PLC con simplesso e gomory in salsa teriyaki}{25}{subsubsection.3.5.2}% +\contentsline {subsubsection}{\numberline {3.5.3}Senstivity Analysis}{27}{subsubsection.3.5.3}% +\contentsline {section}{\numberline {4}Grafi}{29}{section.4}% +\contentsline {subsection}{\numberline {4.1}GT}{29}{subsection.4.1}% +\contentsline {subsubsection}{\numberline {4.1.1}Dijkstra}{29}{subsubsection.4.1.1}% +\contentsline {subsubsection}{\numberline {4.1.2}Shortest Path Tree}{31}{subsubsection.4.1.2}% +\contentsline {subsection}{\numberline {4.2}SST}{32}{subsection.4.2}% +\contentsline {subsubsection}{\numberline {4.2.1}Cenni di Base}{32}{subsubsection.4.2.1}% +\contentsline {subsubsection}{\numberline {4.2.2}SST Prim's}{33}{subsubsection.4.2.2}% +\contentsline {subsubsection}{\numberline {4.2.3}SST Da Matrice trovare la soluzione ottimale}{34}{subsubsection.4.2.3}% +\contentsline {subsection}{\numberline {4.3}Max Flow}{35}{subsection.4.3}% +\contentsline {subsubsection}{\numberline {4.3.1}Cenni di Base}{35}{subsubsection.4.3.1}% +\contentsline {subsubsection}{\numberline {4.3.2}Flow Network}{36}{subsubsection.4.3.2}% +\contentsline {subsection}{\numberline {4.4}DP}{38}{subsection.4.4}% +\contentsline {subsubsection}{\numberline {4.4.1}DP Knapsack 0-1 Dynamic Programming}{38}{subsubsection.4.4.1}% +\contentsline {subsubsection}{\numberline {4.4.2}DP: SSP Bellman's-Ford}{40}{subsubsection.4.4.2}% +\contentsline {subsection}{\numberline {4.5}ILP}{42}{subsection.4.5}% +\contentsline {subsubsection}{\numberline {4.5.1}ILP standard B\&B}{42}{subsubsection.4.5.1}% +\contentsline {subsubsection}{\numberline {4.5.2}ILP esercizio standard B\&B}{46}{subsubsection.4.5.2}% +\contentsline {subsubsection}{\numberline {4.5.3}ILP esercizio B\&B 0-1}{48}{subsubsection.4.5.3}% +\contentsline {section}{\numberline {5}GPLK}{50}{section.5}% +\contentsline {subsection}{\numberline {5.1}Dal modello al codice}{50}{subsection.5.1}% +\contentsline {subsubsection}{\numberline {5.1.1}Modello Easy}{50}{subsubsection.5.1.1}% +\contentsline {subsubsection}{\numberline {5.1.2}Caso Particolare 1 Graffa}{51}{subsubsection.5.1.2}% +\contentsline {subsubsection}{\numberline {5.1.3}Caso Particolare 2 Sommatoria doppio insieme}{53}{subsubsection.5.1.3}% +\contentsline {subsection}{\numberline {5.2}Dal codice al modello}{54}{subsection.5.2}% +\contentsline {subsubsection}{\numberline {5.2.1}Normale}{54}{subsubsection.5.2.1}% +\contentsline {section}{\numberline {6}Domande varie di teoria}{55}{section.6}% +\contentsline {subsection}{\numberline {6.1}PLC degenerate}{55}{subsection.6.1}% +\contentsline {subsection}{\numberline {6.2}PLC sensitivty}{56}{subsection.6.2}% +\contentsline {subsection}{\numberline {6.3}B\&B}{56}{subsection.6.3}% +\contentsline {subsection}{\numberline {6.4}PLC minimization}{56}{subsection.6.4}% +\contentsline {subsection}{\numberline {6.5}cutting Plane}{56}{subsection.6.5}% +\contentsline {subsection}{\numberline {6.6}PLC dual}{57}{subsection.6.6}% +\contentsline {subsection}{\numberline {6.7}Dijkstra}{57}{subsection.6.7}% +\contentsline {subsection}{\numberline {6.8}B\&B knapsack}{57}{subsection.6.8}% +\contentsline {subsection}{\numberline {6.9}Branch \& cut}{57}{subsection.6.9}% +\contentsline {subsection}{\numberline {6.10}PLC}{57}{subsection.6.10}%