sadasd

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{vmargin}
\setpapersize{A4}


\title{Laboratorio \#2: Optimización\\
Investigación de Operaciones 1}
\author{Alumnos: Dante Garin Velásquez - Felipe Silva Meneses\\
\\
Ayudantes: Claudia Chacón Ossa - Camila Díaz Cifuentes\\
\\
Profesor: Alvaro Luzzi Haussmann\\}
\date{Diciembre 2015}

\usepackage{natbib}
\usepackage{graphicx}
\usepackage{multirow}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{tikz}
\usepackage{tkz-berge}
\begin{document}


\maketitle
\part{Solución Pregunta 1}
Declaración de variables:\\
Sea \begin{equation*}
X_{i,j} = \begin{cases}
1 &\mbox{Si el jugador $i$ se encuentra presente en el juego $j$ }\\
0 &\mbox{Si no}\\
\end{cases}
\end{equation*}
\\
Parámetros:
\begin{equation*}
\mbox{P}_{i,j}:\mbox{Puntaje del jugador $i$ en el juego $j$}
\end{equation*}
\\
Función Objetivo:
\begin{equation*}
\mbox{Max  }z=\sum\limits_{i=1}^6 \sum\limits_{j=1}^2 \mbox{P}_{i,j}\cdot X_{i,j}
\end{equation*}
\\
Restricciones:
\begin{equation}
\sum\limits_{i=1}^6 X_{i,1} = 4 \mbox{\- (4 Participantes para piso).}
\end{equation}
\begin{equation}
\sum\limits_{i=1}^6 X_{i,2} = 4 \mbox{\- (4 Participantes para aparatos).}\\
\end{equation}
\begin{equation*}
X_{i,j} = \{0,1\}
\end{equation*}
\begin{equation*}
\forall \mbox{ }i\mbox{    } \in \mathbb{N}^{+} [ 1,6 ]
\end{equation*}
\begin{equation}
\forall \mbox{ }j\mbox{    }\in \{1,2\}\\
\end{equation}
\begin{equation*}
j = \begin{cases}
1 : &\mbox{Aparatos}\\
2 : &\mbox{Piso}\\
\end{cases}
\end{equation*}


\newpage
\maketitle
\part{Solución Pregunta 2}

\newpage
\maketitle
\part{Solución Pregunta 3}

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\setcounter{equation}{0}
\begin{equation}
C_{E1,4}=(12-K_1,0+K_1)=(0,12)
\end{equation}
\begin{align*}
C_{4,7}=(13-K_1,0+K_1)&=(1,12)\\
C_{7,10}=(20-K_1,0+K_1)&=(8,12)\\
C_{10,P2}=(14-K_1,0+K_1)&=(2,12)\\
E1 - 4 - 7 - 10 - P2 \longrightarrow  K_1 = 12\\
\end{align*}

\begin{equation}
C_{E2,5}=(15-K_2,0+K_2)=(8,7)
\end{equation}
\begin{align*}
C_{5,8}=(23-K_2,0+K_2)&=(16,7)\\
C_{8,10}=(10-K_2,0+K_2)&=(3,7)\\
C_{10,P3}=(7-K_2,0+K_2)&=(0,7)\\
E2 - 5 - 8 - 10 - P3 \longrightarrow  K_2 = 7\\
\end{align*}

\begin{equation}
C_{E2,5}=(8-K_3,7+K_3)=(0,15)
\end{equation}
\begin{align*}
C_{5,8}=(16-K_3,7+K_3)&=(8,15)\\
C_{8,P3}=(8-K_3,0+K_3)&=(0,8)\\
E2 - 5 - 8  - P3 \longrightarrow  K_3 = 8\\
\end{align*}

\begin{equation}
C_{E2,4}=(5-K_4,0+K_4)=(0,5)
\end{equation}
\begin{align*}
C_{4,6}=(7-K_4,0+K_4)&=(2,5)\\
C_{6,P1}=(25-K_4,0+K_4)&=(20,5)\\
E2 - 4 - 6  - P1 \longrightarrow  K_4 = 5\\
\end{align*}

\begin{equation}
C_{E3,5}=(20-K_5,0+K_5)=(18,2)
\end{equation}
\begin{align*}
C_{5,7}=(16-K_5,0+K_5)&=(14,2)\\
C_{7,10}=(8-K_5,12+K_5)&=(6,14)\\
C_{10,P2}=(2-K_5,12+K_5)&=(0,14)\\
E3 - 5 - 7 - 10 - P2 \longrightarrow  K_5 = 2\\
\\
\end{align*}
\begin{align*}
\mbox{Flujo Máximo } = \sum\limits_{k=1}^5 k = 12+7+8+5+2 = 34
\end{align*}


\end{document}