Commit 1b1c720b by Theo von Arx

### 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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