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V1.0a
Aggiunte risposte domande di teoria sezione 6 Hyperlink nel nome dell'autore che riporta al github
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meschio94
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7 changed files with 849 additions and 568 deletions
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|
\BOOKMARK [2][-]{subsection.4.2}{SST}{section.4}% 33
|
||||||
|
\BOOKMARK [3][-]{subsubsection.4.2.1}{Cenni di Base}{subsection.4.2}% 34
|
||||||
|
\BOOKMARK [3][-]{subsubsection.4.2.2}{SST Prim's}{subsection.4.2}% 35
|
||||||
|
\BOOKMARK [3][-]{subsubsection.4.2.3}{SST Da Matrice trovare la soluzione ottimale}{subsection.4.2}% 36
|
||||||
|
\BOOKMARK [2][-]{subsection.4.3}{Max Flow}{section.4}% 37
|
||||||
|
\BOOKMARK [3][-]{subsubsection.4.3.1}{Cenni di Base}{subsection.4.3}% 38
|
||||||
|
\BOOKMARK [3][-]{subsubsection.4.3.2}{Flow Network}{subsection.4.3}% 39
|
||||||
|
\BOOKMARK [2][-]{subsection.4.4}{DP}{section.4}% 40
|
||||||
|
\BOOKMARK [3][-]{subsubsection.4.4.1}{DP Knapsack 0-1 Dynamic Programming}{subsection.4.4}% 41
|
||||||
|
\BOOKMARK [3][-]{subsubsection.4.4.2}{DP: SSP Bellman's-Ford}{subsection.4.4}% 42
|
||||||
|
\BOOKMARK [2][-]{subsection.4.5}{ILP}{section.4}% 43
|
||||||
|
\BOOKMARK [3][-]{subsubsection.4.5.1}{ILP standard B\046B}{subsection.4.5}% 44
|
||||||
|
\BOOKMARK [3][-]{subsubsection.4.5.2}{ILP esercizio standard B\046B}{subsection.4.5}% 45
|
||||||
|
\BOOKMARK [3][-]{subsubsection.4.5.3}{ILP esercizio B\046B 0-1}{subsection.4.5}% 46
|
||||||
|
\BOOKMARK [1][-]{section.5}{GPLK}{}% 47
|
||||||
|
\BOOKMARK [2][-]{subsection.5.1}{Dal modello al codice}{section.5}% 48
|
||||||
|
\BOOKMARK [3][-]{subsubsection.5.1.1}{Modello Easy}{subsection.5.1}% 49
|
||||||
|
\BOOKMARK [3][-]{subsubsection.5.1.2}{Caso Particolare 1 Graffa}{subsection.5.1}% 50
|
||||||
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\BOOKMARK [3][-]{subsubsection.5.1.3}{Caso Particolare 2 Sommatoria doppio insieme}{subsection.5.1}% 51
|
||||||
|
\BOOKMARK [2][-]{subsection.5.2}{Dal codice al modello}{section.5}% 52
|
||||||
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\BOOKMARK [3][-]{subsubsection.5.2.1}{Normale}{subsection.5.2}% 53
|
||||||
|
\BOOKMARK [1][-]{section.6}{Domande varie di teoria}{}% 54
|
||||||
|
\BOOKMARK [2][-]{subsection.6.1}{PLC degenerate}{section.6}% 55
|
||||||
|
\BOOKMARK [2][-]{subsection.6.2}{PLC sensitivty}{section.6}% 56
|
||||||
|
\BOOKMARK [2][-]{subsection.6.3}{B\046B}{section.6}% 57
|
||||||
|
\BOOKMARK [2][-]{subsection.6.4}{PLC minimization}{section.6}% 58
|
||||||
|
\BOOKMARK [2][-]{subsection.6.5}{cutting Plane}{section.6}% 59
|
||||||
|
\BOOKMARK [2][-]{subsection.6.6}{PLC dual}{section.6}% 60
|
||||||
|
\BOOKMARK [2][-]{subsection.6.7}{Dijkstra}{section.6}% 61
|
||||||
|
\BOOKMARK [2][-]{subsection.6.8}{B\046B knapsack}{section.6}% 62
|
||||||
|
\BOOKMARK [2][-]{subsection.6.9}{Branch \046 cut}{section.6}% 63
|
||||||
|
\BOOKMARK [2][-]{subsection.6.10}{PLC}{section.6}% 64
|
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@ -3,21 +3,23 @@
|
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%Headers
|
%Headers
|
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\usepackage[dvips]{graphicx} %package that does pdfs
|
\usepackage[dvips]{graphicx} %package that does pdfs
|
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\usepackage{color} %this needs to be here also
|
\usepackage{color} %this needs to be here also
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\usepackage{amssymb}
|
\usepackage{amssymb} %librerie simboli matematica
|
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\usepackage{amsmath}
|
\usepackage{amsmath}
|
||||||
|
|
||||||
\usepackage{xcolor,listings}
|
\usepackage{xcolor,listings} %per il codice
|
||||||
\usepackage{textcomp}
|
\usepackage{textcomp}
|
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\lstset{upquote=true}
|
\lstset{upquote=true}
|
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|
|
||||||
\usepackage{imakeidx}
|
\usepackage{hyperref} %per gli hyperlink
|
||||||
|
|
||||||
|
\usepackage{imakeidx} %per l'indice
|
||||||
\makeindex
|
\makeindex
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
\title{Gran Compendio OLI}
|
\title{Gran Compendio OLI}
|
||||||
\author{Meschio}
|
\author{\href{https://github.com/meschio94/Gran-Compendio-OLI}{Meschio}}
|
||||||
\date{2021\\V1.0}
|
\date{2021\\V1.0a}
|
||||||
|
|
||||||
\begin{document}
|
\begin{document}
|
||||||
\maketitle
|
\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.
|
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}
|
\subsection{Dijkstra}
|
||||||
\noindent\fbox{%
|
\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.
|
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}
|
\subsection{B\&B knapsack}
|
||||||
\noindent\fbox{%
|
\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.
|
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}
|
\subsection{Branch \& cut}
|
||||||
\noindent\fbox{%
|
\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.
|
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}
|
\subsection{PLC}
|
||||||
\noindent\fbox{%
|
\noindent\fbox{%
|
||||||
\parbox{\textwidth}{%
|
\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.
|
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}
|
\end{document}
|
||||||
|
|
|
||||||
|
|
@ -1,64 +1,64 @@
|
||||||
\contentsline {section}{\numberline {1}Modelli}{3}{}%
|
\contentsline {section}{\numberline {1}Modelli}{1}{section.1}%
|
||||||
\contentsline {subsection}{\numberline {1.1}Cenni di Base}{3}{}%
|
\contentsline {subsection}{\numberline {1.1}Cenni di Base}{1}{subsection.1.1}%
|
||||||
\contentsline {subsection}{\numberline {1.2}Esercizi}{5}{}%
|
\contentsline {subsection}{\numberline {1.2}Esercizi}{4}{subsection.1.2}%
|
||||||
\contentsline {subsubsection}{\numberline {1.2.1}Problema con Delta}{5}{}%
|
\contentsline {subsubsection}{\numberline {1.2.1}Problema con Delta}{4}{subsubsection.1.2.1}%
|
||||||
\contentsline {subsubsection}{\numberline {1.2.2}Problema con massima distanza}{6}{}%
|
\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}{7}{}%
|
\contentsline {subsubsection}{\numberline {1.2.3}Problema Variabile Triplo indice}{6}{subsubsection.1.2.3}%
|
||||||
\contentsline {subsubsection}{\numberline {1.2.4}Problema di Trasporto}{9}{}%
|
\contentsline {subsubsection}{\numberline {1.2.4}Problema di Trasporto}{8}{subsubsection.1.2.4}%
|
||||||
\contentsline {subsubsection}{\numberline {1.2.5}Problema di Stoccaggio}{11}{}%
|
\contentsline {subsubsection}{\numberline {1.2.5}Problema di Stoccaggio}{10}{subsubsection.1.2.5}%
|
||||||
\contentsline {subsubsection}{\numberline {1.2.6}Problema di Stoccaggio 2}{12}{}%
|
\contentsline {subsubsection}{\numberline {1.2.6}Problema di Stoccaggio 2}{11}{subsubsection.1.2.6}%
|
||||||
\contentsline {section}{\numberline {2}Forme: Standard, Canonica, Generale}{13}{}%
|
\contentsline {section}{\numberline {2}Forme: Standard, Canonica, Generale}{12}{section.2}%
|
||||||
\contentsline {subsection}{\numberline {2.1}Standard $\Rightarrow $ Canonica}{13}{}%
|
\contentsline {subsection}{\numberline {2.1}Standard $\Rightarrow $ Canonica}{12}{subsection.2.1}%
|
||||||
\contentsline {subsection}{\numberline {2.2}Generale $\Rightarrow $ Standard}{13}{}%
|
\contentsline {subsection}{\numberline {2.2}Generale $\Rightarrow $ Standard}{12}{subsection.2.2}%
|
||||||
\contentsline {subsection}{\numberline {2.3}Esercizio Trasformazione Gran Fritto Misto}{14}{}%
|
\contentsline {subsection}{\numberline {2.3}Esercizio Trasformazione Gran Fritto Misto}{13}{subsection.2.3}%
|
||||||
\contentsline {section}{\numberline {3}Matrici : LP, ILP}{15}{}%
|
\contentsline {section}{\numberline {3}Matrici : LP, ILP}{14}{section.3}%
|
||||||
\contentsline {subsection}{\numberline {3.1}Fondamenti Concettuali}{15}{}%
|
\contentsline {subsection}{\numberline {3.1}Fondamenti Concettuali}{14}{subsection.3.1}%
|
||||||
\contentsline {subsubsection}{\numberline {3.1.1}Quale Simplesso?}{15}{}%
|
\contentsline {subsubsection}{\numberline {3.1.1}Quale Simplesso?}{14}{subsubsection.3.1.1}%
|
||||||
\contentsline {subsection}{\numberline {3.2}LP}{15}{}%
|
\contentsline {subsection}{\numberline {3.2}LP}{14}{subsection.3.2}%
|
||||||
\contentsline {subsubsection}{\numberline {3.2.1}Simplesso}{15}{}%
|
\contentsline {subsubsection}{\numberline {3.2.1}Simplesso}{14}{subsubsection.3.2.1}%
|
||||||
\contentsline {subsubsection}{\numberline {3.2.2}Simplesso Duale}{16}{}%
|
\contentsline {subsubsection}{\numberline {3.2.2}Simplesso Duale}{15}{subsubsection.3.2.2}%
|
||||||
\contentsline {subsubsection}{\numberline {3.2.3}2Fasi}{17}{}%
|
\contentsline {subsubsection}{\numberline {3.2.3}2Fasi}{16}{subsubsection.3.2.3}%
|
||||||
\contentsline {subsubsection}{\numberline {3.2.4}Duale del problema}{19}{}%
|
\contentsline {subsubsection}{\numberline {3.2.4}Duale del problema}{18}{subsubsection.3.2.4}%
|
||||||
\contentsline {subsection}{\numberline {3.3}Rappresentazione Grafica}{20}{}%
|
\contentsline {subsection}{\numberline {3.3}Rappresentazione Grafica}{19}{subsection.3.3}%
|
||||||
\contentsline {subsection}{\numberline {3.4}ILP}{22}{}%
|
\contentsline {subsection}{\numberline {3.4}ILP}{21}{subsection.3.4}%
|
||||||
\contentsline {subsubsection}{\numberline {3.4.1}Tagli di Gomory}{22}{}%
|
\contentsline {subsubsection}{\numberline {3.4.1}Tagli di Gomory}{21}{subsubsection.3.4.1}%
|
||||||
\contentsline {subsection}{\numberline {3.5}Esercizi Particolari}{24}{}%
|
\contentsline {subsection}{\numberline {3.5}Esercizi Particolari}{23}{subsection.3.5}%
|
||||||
\contentsline {subsubsection}{\numberline {3.5.1}Simplesso con variabili free}{24}{}%
|
\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}{26}{}%
|
\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}{28}{}%
|
\contentsline {subsubsection}{\numberline {3.5.3}Senstivity Analysis}{27}{subsubsection.3.5.3}%
|
||||||
\contentsline {section}{\numberline {4}Grafi}{30}{}%
|
\contentsline {section}{\numberline {4}Grafi}{29}{section.4}%
|
||||||
\contentsline {subsection}{\numberline {4.1}GT}{30}{}%
|
\contentsline {subsection}{\numberline {4.1}GT}{29}{subsection.4.1}%
|
||||||
\contentsline {subsubsection}{\numberline {4.1.1}Dijkstra}{30}{}%
|
\contentsline {subsubsection}{\numberline {4.1.1}Dijkstra}{29}{subsubsection.4.1.1}%
|
||||||
\contentsline {subsubsection}{\numberline {4.1.2}Shortest Path Tree}{32}{}%
|
\contentsline {subsubsection}{\numberline {4.1.2}Shortest Path Tree}{31}{subsubsection.4.1.2}%
|
||||||
\contentsline {subsection}{\numberline {4.2}SST}{33}{}%
|
\contentsline {subsection}{\numberline {4.2}SST}{32}{subsection.4.2}%
|
||||||
\contentsline {subsubsection}{\numberline {4.2.1}Cenni di Base}{33}{}%
|
\contentsline {subsubsection}{\numberline {4.2.1}Cenni di Base}{32}{subsubsection.4.2.1}%
|
||||||
\contentsline {subsubsection}{\numberline {4.2.2}SST Prim's}{34}{}%
|
\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}{35}{}%
|
\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}{36}{}%
|
\contentsline {subsection}{\numberline {4.3}Max Flow}{35}{subsection.4.3}%
|
||||||
\contentsline {subsubsection}{\numberline {4.3.1}Cenni di Base}{36}{}%
|
\contentsline {subsubsection}{\numberline {4.3.1}Cenni di Base}{35}{subsubsection.4.3.1}%
|
||||||
\contentsline {subsubsection}{\numberline {4.3.2}Flow Network}{37}{}%
|
\contentsline {subsubsection}{\numberline {4.3.2}Flow Network}{36}{subsubsection.4.3.2}%
|
||||||
\contentsline {subsection}{\numberline {4.4}DP}{39}{}%
|
\contentsline {subsection}{\numberline {4.4}DP}{38}{subsection.4.4}%
|
||||||
\contentsline {subsubsection}{\numberline {4.4.1}DP Knapsack 0-1 Dynamic Programming}{39}{}%
|
\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}{41}{}%
|
\contentsline {subsubsection}{\numberline {4.4.2}DP: SSP Bellman's-Ford}{40}{subsubsection.4.4.2}%
|
||||||
\contentsline {subsection}{\numberline {4.5}ILP}{43}{}%
|
\contentsline {subsection}{\numberline {4.5}ILP}{42}{subsection.4.5}%
|
||||||
\contentsline {subsubsection}{\numberline {4.5.1}ILP standard B\&B}{43}{}%
|
\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}{47}{}%
|
\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}{49}{}%
|
\contentsline {subsubsection}{\numberline {4.5.3}ILP esercizio B\&B 0-1}{48}{subsubsection.4.5.3}%
|
||||||
\contentsline {section}{\numberline {5}GPLK}{51}{}%
|
\contentsline {section}{\numberline {5}GPLK}{50}{section.5}%
|
||||||
\contentsline {subsection}{\numberline {5.1}Dal modello al codice}{51}{}%
|
\contentsline {subsection}{\numberline {5.1}Dal modello al codice}{50}{subsection.5.1}%
|
||||||
\contentsline {subsubsection}{\numberline {5.1.1}Modello Easy}{51}{}%
|
\contentsline {subsubsection}{\numberline {5.1.1}Modello Easy}{50}{subsubsection.5.1.1}%
|
||||||
\contentsline {subsubsection}{\numberline {5.1.2}Caso Particolare 1 Graffa}{52}{}%
|
\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}{54}{}%
|
\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}{55}{}%
|
\contentsline {subsection}{\numberline {5.2}Dal codice al modello}{54}{subsection.5.2}%
|
||||||
\contentsline {subsubsection}{\numberline {5.2.1}Normale}{55}{}%
|
\contentsline {subsubsection}{\numberline {5.2.1}Normale}{54}{subsubsection.5.2.1}%
|
||||||
\contentsline {section}{\numberline {6}Domande varie di teoria}{56}{}%
|
\contentsline {section}{\numberline {6}Domande varie di teoria}{55}{section.6}%
|
||||||
\contentsline {subsection}{\numberline {6.1}PLC degenerate}{56}{}%
|
\contentsline {subsection}{\numberline {6.1}PLC degenerate}{55}{subsection.6.1}%
|
||||||
\contentsline {subsection}{\numberline {6.2}PLC sensitivty}{57}{}%
|
\contentsline {subsection}{\numberline {6.2}PLC sensitivty}{56}{subsection.6.2}%
|
||||||
\contentsline {subsection}{\numberline {6.3}B\&B}{57}{}%
|
\contentsline {subsection}{\numberline {6.3}B\&B}{56}{subsection.6.3}%
|
||||||
\contentsline {subsection}{\numberline {6.4}PLC minimization}{57}{}%
|
\contentsline {subsection}{\numberline {6.4}PLC minimization}{56}{subsection.6.4}%
|
||||||
\contentsline {subsection}{\numberline {6.5}cutting Plane}{57}{}%
|
\contentsline {subsection}{\numberline {6.5}cutting Plane}{56}{subsection.6.5}%
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\contentsline {subsection}{\numberline {6.6}PLC dual}{58}{}%
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\contentsline {subsection}{\numberline {6.6}PLC dual}{57}{subsection.6.6}%
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\contentsline {subsection}{\numberline {6.7}Dijkstra}{58}{}%
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\contentsline {subsection}{\numberline {6.7}Dijkstra}{57}{subsection.6.7}%
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\contentsline {subsection}{\numberline {6.8}B\&B knapsack}{58}{}%
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\contentsline {subsection}{\numberline {6.8}B\&B knapsack}{57}{subsection.6.8}%
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\contentsline {subsection}{\numberline {6.9}Branch \& cut}{58}{}%
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\contentsline {subsection}{\numberline {6.9}Branch \& cut}{57}{subsection.6.9}%
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\contentsline {subsection}{\numberline {6.10}PLC}{58}{}%
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\contentsline {subsection}{\numberline {6.10}PLC}{57}{subsection.6.10}%
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Reference in a new issue