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fix sezione 6

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meschio94 2021-06-15 13:42:35 +02:00
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commit 019145cf87
6 changed files with 284 additions and 286 deletions
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@ -1360,6 +1360,7 @@ 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. 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:\\ Soluzione:\\
(A,B,D) (A,B,D)
@ -1369,6 +1370,7 @@ Soluzione:\\
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. 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.
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Soluzione:\\ Soluzione:\\
(A,B,C) (A,B,C)
@ -1378,6 +1380,7 @@ Soluzione:\\
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. 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.
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Soluzione:\\ Soluzione:\\
(B,E) (B,E)
@ -1387,6 +1390,7 @@ Soluzione:\\
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. 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.
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Soluzione:\\ Soluzione:\\
(C,E) (C,E)
\subsection{PLC} \subsection{PLC}
@ -1395,6 +1399,7 @@ Soluzione:\\
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. 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.
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Soluzione:\\ Soluzione:\\
(B) (B)
\end{document} \end{document}

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\contentsline {section}{\numberline {1}Modelli}{1}{section.1}% \contentsline {section}{\numberline {1}Modelli}{3}{section.1}%
\contentsline {subsection}{\numberline {1.1}Cenni di Base}{1}{subsection.1.1}% \contentsline {subsection}{\numberline {1.1}Cenni di Base}{3}{subsection.1.1}%
\contentsline {subsection}{\numberline {1.2}Esercizi}{4}{subsection.1.2}% \contentsline {subsection}{\numberline {1.2}Esercizi}{5}{subsection.1.2}%
\contentsline {subsubsection}{\numberline {1.2.1}Problema con Delta}{4}{subsubsection.1.2.1}% \contentsline {subsubsection}{\numberline {1.2.1}Problema con Delta}{5}{subsubsection.1.2.1}%
\contentsline {subsubsection}{\numberline {1.2.2}Problema con massima distanza}{5}{subsubsection.1.2.2}% \contentsline {subsubsection}{\numberline {1.2.2}Problema con massima distanza}{6}{subsubsection.1.2.2}%
\contentsline {subsubsection}{\numberline {1.2.3}Problema Variabile Triplo indice}{6}{subsubsection.1.2.3}% \contentsline {subsubsection}{\numberline {1.2.3}Problema Variabile Triplo indice}{7}{subsubsection.1.2.3}%
\contentsline {subsubsection}{\numberline {1.2.4}Problema di Trasporto}{8}{subsubsection.1.2.4}% \contentsline {subsubsection}{\numberline {1.2.4}Problema di Trasporto}{9}{subsubsection.1.2.4}%
\contentsline {subsubsection}{\numberline {1.2.5}Problema di Stoccaggio}{10}{subsubsection.1.2.5}% \contentsline {subsubsection}{\numberline {1.2.5}Problema di Stoccaggio}{11}{subsubsection.1.2.5}%
\contentsline {subsubsection}{\numberline {1.2.6}Problema di Stoccaggio 2}{11}{subsubsection.1.2.6}% \contentsline {subsubsection}{\numberline {1.2.6}Problema di Stoccaggio 2}{12}{subsubsection.1.2.6}%
\contentsline {section}{\numberline {2}Forme: Standard, Canonica, Generale}{12}{section.2}% \contentsline {section}{\numberline {2}Forme: Standard, Canonica, Generale}{13}{section.2}%
\contentsline {subsection}{\numberline {2.1}Standard $\Rightarrow $ Canonica}{12}{subsection.2.1}% \contentsline {subsection}{\numberline {2.1}Standard $\Rightarrow $ Canonica}{13}{subsection.2.1}%
\contentsline {subsection}{\numberline {2.2}Generale $\Rightarrow $ Standard}{12}{subsection.2.2}% \contentsline {subsection}{\numberline {2.2}Generale $\Rightarrow $ Standard}{13}{subsection.2.2}%
\contentsline {subsection}{\numberline {2.3}Esercizio Trasformazione Gran Fritto Misto}{13}{subsection.2.3}% \contentsline {subsection}{\numberline {2.3}Esercizio Trasformazione Gran Fritto Misto}{14}{subsection.2.3}%
\contentsline {section}{\numberline {3}Matrici : LP, ILP}{14}{section.3}% \contentsline {section}{\numberline {3}Matrici : LP, ILP}{15}{section.3}%
\contentsline {subsection}{\numberline {3.1}Fondamenti Concettuali}{14}{subsection.3.1}% \contentsline {subsection}{\numberline {3.1}Fondamenti Concettuali}{15}{subsection.3.1}%
\contentsline {subsubsection}{\numberline {3.1.1}Quale Simplesso?}{14}{subsubsection.3.1.1}% \contentsline {subsubsection}{\numberline {3.1.1}Quale Simplesso?}{15}{subsubsection.3.1.1}%
\contentsline {subsection}{\numberline {3.2}LP}{14}{subsection.3.2}% \contentsline {subsection}{\numberline {3.2}LP}{15}{subsection.3.2}%
\contentsline {subsubsection}{\numberline {3.2.1}Simplesso}{14}{subsubsection.3.2.1}% \contentsline {subsubsection}{\numberline {3.2.1}Simplesso}{15}{subsubsection.3.2.1}%
\contentsline {subsubsection}{\numberline {3.2.2}Simplesso Duale}{15}{subsubsection.3.2.2}% \contentsline {subsubsection}{\numberline {3.2.2}Simplesso Duale}{16}{subsubsection.3.2.2}%
\contentsline {subsubsection}{\numberline {3.2.3}2Fasi}{16}{subsubsection.3.2.3}% \contentsline {subsubsection}{\numberline {3.2.3}2Fasi}{17}{subsubsection.3.2.3}%
\contentsline {subsubsection}{\numberline {3.2.4}Duale del problema}{18}{subsubsection.3.2.4}% \contentsline {subsubsection}{\numberline {3.2.4}Duale del problema}{19}{subsubsection.3.2.4}%
\contentsline {subsection}{\numberline {3.3}Rappresentazione Grafica}{19}{subsection.3.3}% \contentsline {subsection}{\numberline {3.3}Rappresentazione Grafica}{20}{subsection.3.3}%
\contentsline {subsection}{\numberline {3.4}ILP}{21}{subsection.3.4}% \contentsline {subsection}{\numberline {3.4}ILP}{22}{subsection.3.4}%
\contentsline {subsubsection}{\numberline {3.4.1}Tagli di Gomory}{21}{subsubsection.3.4.1}% \contentsline {subsubsection}{\numberline {3.4.1}Tagli di Gomory}{22}{subsubsection.3.4.1}%
\contentsline {subsection}{\numberline {3.5}Esercizi Particolari}{23}{subsection.3.5}% \contentsline {subsection}{\numberline {3.5}Esercizi Particolari}{24}{subsection.3.5}%
\contentsline {subsubsection}{\numberline {3.5.1}Simplesso con variabili free}{23}{subsubsection.3.5.1}% \contentsline {subsubsection}{\numberline {3.5.1}Simplesso con variabili free}{24}{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 {3.5.2}Problema PLC con simplesso e gomory in salsa teriyaki}{26}{subsubsection.3.5.2}%
\contentsline {subsubsection}{\numberline {3.5.3}Senstivity Analysis}{27}{subsubsection.3.5.3}% \contentsline {subsubsection}{\numberline {3.5.3}Senstivity Analysis}{28}{subsubsection.3.5.3}%
\contentsline {section}{\numberline {4}Grafi}{29}{section.4}% \contentsline {section}{\numberline {4}Grafi}{30}{section.4}%
\contentsline {subsection}{\numberline {4.1}GT}{29}{subsection.4.1}% \contentsline {subsection}{\numberline {4.1}GT}{30}{subsection.4.1}%
\contentsline {subsubsection}{\numberline {4.1.1}Dijkstra}{29}{subsubsection.4.1.1}% \contentsline {subsubsection}{\numberline {4.1.1}Dijkstra}{30}{subsubsection.4.1.1}%
\contentsline {subsubsection}{\numberline {4.1.2}Shortest Path Tree}{31}{subsubsection.4.1.2}% \contentsline {subsubsection}{\numberline {4.1.2}Shortest Path Tree}{32}{subsubsection.4.1.2}%
\contentsline {subsection}{\numberline {4.2}SST}{32}{subsection.4.2}% \contentsline {subsection}{\numberline {4.2}SST}{33}{subsection.4.2}%
\contentsline {subsubsection}{\numberline {4.2.1}Cenni di Base}{32}{subsubsection.4.2.1}% \contentsline {subsubsection}{\numberline {4.2.1}Cenni di Base}{33}{subsubsection.4.2.1}%
\contentsline {subsubsection}{\numberline {4.2.2}SST Prim's}{33}{subsubsection.4.2.2}% \contentsline {subsubsection}{\numberline {4.2.2}SST Prim's}{34}{subsubsection.4.2.2}%
\contentsline {subsubsection}{\numberline {4.2.3}SST Da Matrice trovare la soluzione ottimale}{34}{subsubsection.4.2.3}% \contentsline {subsubsection}{\numberline {4.2.3}SST Da Matrice trovare la soluzione ottimale}{35}{subsubsection.4.2.3}%
\contentsline {subsection}{\numberline {4.3}Max Flow}{35}{subsection.4.3}% \contentsline {subsection}{\numberline {4.3}Max Flow}{36}{subsection.4.3}%
\contentsline {subsubsection}{\numberline {4.3.1}Cenni di Base}{35}{subsubsection.4.3.1}% \contentsline {subsubsection}{\numberline {4.3.1}Cenni di Base}{36}{subsubsection.4.3.1}%
\contentsline {subsubsection}{\numberline {4.3.2}Flow Network}{36}{subsubsection.4.3.2}% \contentsline {subsubsection}{\numberline {4.3.2}Flow Network}{37}{subsubsection.4.3.2}%
\contentsline {subsection}{\numberline {4.4}DP}{38}{subsection.4.4}% \contentsline {subsection}{\numberline {4.4}DP}{39}{subsection.4.4}%
\contentsline {subsubsection}{\numberline {4.4.1}DP Knapsack 0-1 Dynamic Programming}{38}{subsubsection.4.4.1}% \contentsline {subsubsection}{\numberline {4.4.1}DP Knapsack 0-1 Dynamic Programming}{39}{subsubsection.4.4.1}%
\contentsline {subsubsection}{\numberline {4.4.2}DP: SSP Bellman's-Ford}{40}{subsubsection.4.4.2}% \contentsline {subsubsection}{\numberline {4.4.2}DP: SSP Bellman's-Ford}{41}{subsubsection.4.4.2}%
\contentsline {subsection}{\numberline {4.5}ILP}{42}{subsection.4.5}% \contentsline {subsection}{\numberline {4.5}ILP}{43}{subsection.4.5}%
\contentsline {subsubsection}{\numberline {4.5.1}ILP standard B\&B}{42}{subsubsection.4.5.1}% \contentsline {subsubsection}{\numberline {4.5.1}ILP standard B\&B}{43}{subsubsection.4.5.1}%
\contentsline {subsubsection}{\numberline {4.5.2}ILP esercizio standard B\&B}{46}{subsubsection.4.5.2}% \contentsline {subsubsection}{\numberline {4.5.2}ILP esercizio standard B\&B}{47}{subsubsection.4.5.2}%
\contentsline {subsubsection}{\numberline {4.5.3}ILP esercizio B\&B 0-1}{48}{subsubsection.4.5.3}% \contentsline {subsubsection}{\numberline {4.5.3}ILP esercizio B\&B 0-1}{49}{subsubsection.4.5.3}%
\contentsline {section}{\numberline {5}GPLK}{50}{section.5}% \contentsline {section}{\numberline {5}GPLK}{51}{section.5}%
\contentsline {subsection}{\numberline {5.1}Dal modello al codice}{50}{subsection.5.1}% \contentsline {subsection}{\numberline {5.1}Dal modello al codice}{51}{subsection.5.1}%
\contentsline {subsubsection}{\numberline {5.1.1}Modello Easy}{50}{subsubsection.5.1.1}% \contentsline {subsubsection}{\numberline {5.1.1}Modello Easy}{51}{subsubsection.5.1.1}%
\contentsline {subsubsection}{\numberline {5.1.2}Caso Particolare 1 Graffa}{51}{subsubsection.5.1.2}% \contentsline {subsubsection}{\numberline {5.1.2}Caso Particolare 1 Graffa}{52}{subsubsection.5.1.2}%
\contentsline {subsubsection}{\numberline {5.1.3}Caso Particolare 2 Sommatoria doppio insieme}{53}{subsubsection.5.1.3}% \contentsline {subsubsection}{\numberline {5.1.3}Caso Particolare 2 Sommatoria doppio insieme}{54}{subsubsection.5.1.3}%
\contentsline {subsection}{\numberline {5.2}Dal codice al modello}{54}{subsection.5.2}% \contentsline {subsection}{\numberline {5.2}Dal codice al modello}{55}{subsection.5.2}%
\contentsline {subsubsection}{\numberline {5.2.1}Normale}{54}{subsubsection.5.2.1}% \contentsline {subsubsection}{\numberline {5.2.1}Normale}{55}{subsubsection.5.2.1}%
\contentsline {section}{\numberline {6}Domande varie di teoria}{55}{section.6}% \contentsline {section}{\numberline {6}Domande varie di teoria}{56}{section.6}%
\contentsline {subsection}{\numberline {6.1}PLC degenerate}{55}{subsection.6.1}% \contentsline {subsection}{\numberline {6.1}PLC degenerate}{56}{subsection.6.1}%
\contentsline {subsection}{\numberline {6.2}PLC sensitivty}{56}{subsection.6.2}% \contentsline {subsection}{\numberline {6.2}PLC sensitivty}{57}{subsection.6.2}%
\contentsline {subsection}{\numberline {6.3}B\&B}{56}{subsection.6.3}% \contentsline {subsection}{\numberline {6.3}B\&B}{57}{subsection.6.3}%
\contentsline {subsection}{\numberline {6.4}PLC minimization}{56}{subsection.6.4}% \contentsline {subsection}{\numberline {6.4}PLC minimization}{57}{subsection.6.4}%
\contentsline {subsection}{\numberline {6.5}cutting Plane}{56}{subsection.6.5}% \contentsline {subsection}{\numberline {6.5}cutting Plane}{57}{subsection.6.5}%
\contentsline {subsection}{\numberline {6.6}PLC dual}{57}{subsection.6.6}% \contentsline {subsection}{\numberline {6.6}PLC dual}{58}{subsection.6.6}%
\contentsline {subsection}{\numberline {6.7}Dijkstra}{57}{subsection.6.7}% \contentsline {subsection}{\numberline {6.7}Dijkstra}{58}{subsection.6.7}%
\contentsline {subsection}{\numberline {6.8}B\&B knapsack}{57}{subsection.6.8}% \contentsline {subsection}{\numberline {6.8}B\&B knapsack}{58}{subsection.6.8}%
\contentsline {subsection}{\numberline {6.9}Branch \& cut}{57}{subsection.6.9}% \contentsline {subsection}{\numberline {6.9}Branch \& cut}{58}{subsection.6.9}%
\contentsline {subsection}{\numberline {6.10}PLC}{57}{subsection.6.10}% \contentsline {subsection}{\numberline {6.10}PLC}{59}{subsection.6.10}%