3.7.101.3. Flow models for $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}$, $\mathrm{\pi \pi \pi \pi }$

FigureΒ 3.7.24 presents flow models for the $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}$ and the $\mathrm{\pi \pi \pi \pi }$ constraints. Blue arcs represent feasible flows respectively corresponding to the solutions $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}$$\left(\beta ©{x}_{1}=2,{x}_{2}=4,{x}_{3}=6\beta ͺ,\beta ©{y}_{1}=2,{y}_{2}=4\beta ͺ\right)$ and $\mathrm{\pi \pi \pi \pi }$$\left(\beta ©{x}_{1}=2,{x}_{2}=4,{x}_{3}=5\beta ͺ,\beta ©{y}_{1}=2,{y}_{2}=4,{y}_{3}=5\beta ͺ\right)$, while pink arcs correspond to arcs that cannot carry any flow if the constraint has a solution. Within the context of the $\mathrm{\pi \pi \pi \pi }$ constraint, the assignment ${x}_{1}=1$ is forbidden since $1\beta \mathrm{\pi \pi \pi }\left({y}_{1}\right)\beta ͺ\mathrm{\pi \pi \pi }\left({y}_{2}\right)\beta ͺ\mathrm{\pi \pi \pi }\left({y}_{3}\right)$. Consequently ${x}_{1}=2$ and, since ${y}_{1}$ is the only variable of $\left\{{y}_{1},{y}_{2},{y}_{3}\right\}$ that can be assigned value 2, the assignment ${y}_{1}=3$ is forbidden. Now since $3\beta \mathrm{\pi \pi \pi }\left({y}_{1}\right)\beta ͺ\mathrm{\pi \pi \pi }\left({y}_{2}\right)\beta ͺ\mathrm{\pi \pi \pi }\left({y}_{3}\right)$ the assignment ${x}_{2}=3$ is also forbidden. Finally ${x}_{3}=6$ is forbidden since $6\beta \mathrm{\pi \pi \pi }\left({y}_{1}\right)\beta ͺ\mathrm{\pi \pi \pi }\left({y}_{2}\right)\beta ͺ\mathrm{\pi \pi \pi }\left({y}_{3}\right)$.

Table 3.7.24. Domains of the variables for the $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}$ constraint of FigureΒ 3.7.24.
$i$$\mathrm{\pi \pi \pi }\left({x}_{i}\right)$$i$$\mathrm{\pi \pi \pi }\left({y}_{i}\right)$
1$\left\{1,2\right\}$1$\left\{2,3\right\}$
2$\left\{3,4\right\}$2$\left\{4,5\right\}$
3$\left\{4,5,6\right\}$
Table 3.7.24. Domains of the variables for the $\mathrm{\pi \pi \pi \pi }$ constraint of FigureΒ 3.7.24.
$i$$\mathrm{\pi \pi \pi }\left({x}_{i}\right)$$i$$\mathrm{\pi \pi \pi }\left({y}_{i}\right)$
1$\left\{1,2\right\}$1$\left\{2,3\right\}$
2$\left\{3,4\right\}$2$\left\{4,5\right\}$
3$\left\{4,5,6\right\}$3$\left\{4,5\right\}$