Verified Commit 1b1c720b authored by Theo von Arx's avatar Theo von Arx
Browse files

Reorder example and formal definition

parent 91650223
......@@ -150,7 +150,25 @@ Produces the following (indep. of priorities)\\
\end{compactitem}
~\newline
Formally:
\begin{compactenum}
\item Minimize $$L=\tau(v_n) - \tau(v_0)$$ subject to the constraints 2-5
\item Decision variables binary: $$\forall v_i \in V_S: \quad \forall t: l_i\leq t \leq h_i: \quad x_{i,t}\in\{0,1\}$$
\item Only one variable non-zero: $$\forall v_i \in V_S: \quad \sum_{t=l_i}^{h_i}x_{i,t}=1$$
\item Relation between variables $x$ and starting times $\tau$: $$\forall v_i
\in V_S: \quad \tau(v_i) = \sum_{t=l_i}^{h_i}t\cdot x_{i,t}$$
\item Guarantee of precedence constraints: $$\forall (v_i,v_j) \in E_S: \quad \tau(v_j)-\tau(v_i) \geq w(v_i)$$
\item Guarantee of resource constraints:
\begin{equation*}
\begin{alignedat}{1}
&\forall v_k \in V_T: \quad \forall t: 1 \leq t \leq \max\{h_i: v_i \in V_S\}: \\
&\sum_{\forall i: (v_i,v_k) \in E_R} \sum_{p'=\max\{0,t-h_i\}}^{\min\{w(v_i)-1,t-l_i\}} x_{i,t-p'} \leq \alpha(v_k)
\end{alignedat}
\end{equation*}
\end{compactenum}~\newline
Example:
\includegraphics[width=0.5\linewidth]{integer_LP_example}
\begin{compactitem}
......@@ -173,21 +191,6 @@ Example:
\subitem if a operation needs more than 1 timeslot, the resource constraints get more complicated.
\end{compactitem}
Formally:
\begin{compactenum}
\item Minimize $$L=\tau(v_n) - \tau(v_0)$$ subject to the constraints 2-5
\item Decision variables binary: $$\forall v_i \in V_S: \quad \forall t: l_i\leq t \leq h_i: \quad x_{i,t}\in\{0,1\}$$
\item Only one variable non-zero: $$\forall v_i \in V_S: \quad \sum_{t=l_i}^{h_i}x_{i,t}=1$$
\item Relation between variables $x$ and starting times $\tau$: $$\forall v_i \in V_S: \quad \sum_{t=l_i}^{h_i}t\cdot x_{i,t} = \tau(v_i)$$
\item Guarantee of precedence constraints: $$\forall (v_i,v_j) \in E_S: \quad \tau(v_j)-\tau(v_i) \geq w(v_i)$$
\item Guarantee of resource constraints:
\begin{equation*}
\begin{alignedat}{1}
&\forall v_k \in V_T: \quad \forall t: 1 \leq t \leq \max\{h_i: v_i \in V_S\}: \\
&\sum_{\forall i: (v_i,v_k) \in E_R} \sum_{p'=\max\{0,t-h_i\}}^{\min\{w(v_i)-1,t-l_i\}} x_{i,t-p'} \leq \alpha(v_k)
\end{alignedat}
\end{equation*}
\end{compactenum}~\newline
\ownsubsection{Iterative Algorithms (10-56)}
......
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