From 04f26c69fca011b823bd68c1c96dee0319251aac Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pelletier?= Date: Sun, 8 Apr 2018 01:01:00 -0400 Subject: [PATCH] =?UTF-8?q?ajout=20des=20contraintes=20=C3=A0=20la=20secti?= =?UTF-8?q?on=20mod=C3=A9lisation?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Rapport_de_session/modelisation.tex | 49 ++++++++++++++++++-- Rapport_de_session/rapport_de_recherche.pdf | Bin 392852 -> 409573 bytes 2 files changed, 46 insertions(+), 3 deletions(-) diff --git a/Rapport_de_session/modelisation.tex b/Rapport_de_session/modelisation.tex index 7df4ac7..e01aab2 100644 --- a/Rapport_de_session/modelisation.tex +++ b/Rapport_de_session/modelisation.tex @@ -10,7 +10,7 @@ \subsubsection{Horaire de travail et demande} \label{sec:horairedemande} -Nous considérons un horaire de travail d'une durée de $J$ jours. Chaque journée est composée de $P$ périodes de travail, d'une durée de $\frac{24}{P}$ heures chacunes. La demande de travail pour la période $p$ de la journée $j$ est définie par $d_{p,j}$. La demande totale pour la durée de l'horaire de travail est définie par l'équation \eqref{eq:dtot}. +Nous considérons une planification d'horaires de travail d'une durée de $J$ jours. Chaque journée est composée de $P$ périodes de travail, d'une durée de $\frac{24}{P}$ heures chacunes. La demande de travail pour la période $p$ de la journée $j$ est définie par $d_{p,j}$. La demande totale pour la durée de l'horaire de travail est définie par l'équation \eqref{eq:dtot}. \begin{align} \label{eq:dtot} @@ -39,7 +39,7 @@ Les employés ont respectivement un salaire horaire de $S^{A}$ et encourent un f \subsection{Variables} \label{sec:variables} -On représente les horaires de travail par deux tableaux de variables booliennes $X^{FT}$ et $X^{PT}$ \eqref{tableauxvariables}. +On représente les horaires de travail par deux tableaux de variables booliennes $\mathbf{X}^{FT}$ et $\mathbf{X}^{PT}$ \eqref{tableauxvariables}. \begin{align} \label{tableauxvariables} @@ -58,7 +58,7 @@ Afin de réduire la taille du problème, la liste des horaires quotidiens valide \subsubsection{Variables} \label{sec:horairesvariables} -On définit $H^{A} \in \mathcal{N}^{J}$, un vecteur de variables booléennes $h_{j}$ formant un horaire d'une durée de $J$ journées pour un employé de type $A \in \lbrace FT, PT \rbrace$. +On définit $\vec{H}^{A} \in \N^{J}$, un vecteur de variables booléennes $h_{j}$ formant un horaire d'une durée de $J$ journées pour un employé de type $A \in \lbrace FT, PT \rbrace$. \subsubsection{Contraintes} \label{sec:horairescontraintes} @@ -90,9 +90,52 @@ L'horaire doit respecter le nombre maximum de jours consécutifs travaillés aut \label{fig:automatemaxconsecutif} \end{figure} +On obtient, pour chaque type de travailleur, une liste de vecteurs à laquelle on ajouter le vecteur nul, représentant la situation où l'employé est exclus de l'horaire. + +\subsubsection{Ajout des périodes} +\label{sec:horaireajoutperiode} + +On considère la matrice des horaires quinzomadaires valides \eqref{eq:horairejourvalide} et la matrice des $n_{hq}^A$ horaires quotidiens valides \eqref{eq:horairequotidienvalide}, on obtient un tableau d'horaires par périodes travaillées valides $\mathbf{V}^A \in \N^{3}$ par le produit tensoriel \eqref{eq:produithoraire} +\begin{align} +\label{eq:horairejourvalide} +\mathbf{H}^A &= \left( \vec{H}^{A} \right)\\ +\label{eq:horairequotidienvalide} + \mathbf{Q}^A &= \begin{bmatrix} + q_{1,1}& \cdots& q_{P,1}\\ + \vdots&\ddots&\vdots\\ + q_{1,n_{hq}^A}&\cdots&q_{P,n_{hq}^A} + \end{bmatrix}\\ + \label{eq:produithoraire} + \mathbf{V}^A &= \mathbf{H}^A \otimes \mathbf{Q}^A + \end{align} + \subsection{Contraintes} \label{sec:contraintes} +Chaque horaire d'employé doit d'abord correspondre à un horaire valide tel qu'établi à la section \ref{sec:horairesvalides}. Il doit donc satisfaire une contrainte \textsc{Tableau} \eqref{eq:contraintetableau}. + +\begin{align} + \label{eq:contraintetableau} + \mathcal{C}_1 &: \textsc{Tableau}\left( \mathbf{V}^A, + \begin{bmatrix}x_{e,1,1}&\cdots&x_{e,1,J}\\ + \vdots&\ddots&\vdots\\ + x_{e,P,1}&\cdots&x_{e,P,J} + \end{bmatrix}\right) & \forall e +\end{align} + +La planification doit aussi satisfaire la demande en employés, en utilisant une contrainte \textsc{Sum}. + +\begin{align} + \mathcal{C}_2 &: \sum_{A}\sum_{e=1}^{E^A} x_{e,p,j} = d_{p,j} & \forall 1 \leq p \leq P, 1 \leq j \leq J +\end{align} + +\subsection{Fonction objectif} +\label{sec:fonctionobjectif} + + + + + diff --git a/Rapport_de_session/rapport_de_recherche.pdf b/Rapport_de_session/rapport_de_recherche.pdf index 23a052986233b3ddcff86d53c5eee7a38f466dcc..7203acffbb1832421a2cbd7d1ea1c57418d9c442 100644 GIT binary patch delta 150075 zcmZsBQ*fX`n{KR0CZ5=~ZQHhOJ71hkGO=xIV%xSgv2CA!cdPbhPgP%ZbyZ*WQ~f?l 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