### 3.7.101.4. Flow model for

FigureΒ 3.7.25 presents a flow model for the constraint. Blue arcs represent a feasible flow corresponding to the solution $\left(\beta ©{x}_{1}=2,{x}_{2}=4,{x}_{4}=\beta ͺ,\beta ©{y}_{1}=2,{y}_{2}=4,{y}_{3}=4\beta ͺ,\beta ©\mathrm{\pi \pi \pi }-1\mathrm{\pi \pi \pi \pi }-0\mathrm{\pi \pi \pi \pi ‘}-1,\mathrm{\pi \pi \pi }-2\mathrm{\pi \pi \pi \pi }-1\mathrm{\pi \pi \pi \pi ‘}-2,\mathrm{\pi \pi \pi }-3\mathrm{\pi \pi \pi \pi }-0\mathrm{\pi \pi \pi \pi ‘}-3,\mathrm{\pi \pi \pi }-4\mathrm{\pi \pi \pi \pi }-2\mathrm{\pi \pi \pi \pi ‘}-3,\mathrm{\pi \pi \pi }-5\mathrm{\pi \pi \pi \pi }-0\mathrm{\pi \pi \pi \pi ‘}-2,\mathrm{\pi \pi \pi }-6\mathrm{\pi \pi \pi \pi }-0\mathrm{\pi \pi \pi \pi ‘}-1\beta ͺ\right)$, while pink arcs correspond to arcs that cannot carry any flow if the constraint has a solution. 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. ${x}_{3}=6$ is forbidden since $6\beta \mathrm{\pi \pi \pi }\left({y}_{1}\right)\beta ͺ\mathrm{\pi \pi \pi }\left({y}_{2}\right)\beta ͺ\mathrm{\pi \pi \pi }\left({y}_{3}\right)$. Finally ${x}_{3}=5$ and ${y}_{3}=5$ are also forbidden since value 4 must be assigned to at least two variables.

##### Table 3.7.25. Domains of the variables and minimum and maximum number of occurrences of each value for the constraint of FigureΒ 3.7.25.
$i$$\mathrm{\pi \pi \pi }\left({x}_{i}\right)$$i$$\mathrm{\pi \pi \pi }\left({y}_{i}\right)$$i$$\left[{\mathrm{\pi \pi \pi \pi }}_{i},{\mathrm{\pi \pi \pi \pi ₯}}_{i}\right]$$i$$\left[{\mathrm{\pi \pi \pi \pi }}_{i},{\mathrm{\pi \pi \pi \pi ₯}}_{i}\right]$
1$\left\{1,2\right\}$1$\left\{2,3\right\}$1$\left[0,1\right]$4$\left[2,3\right]$
2$\left\{3,4\right\}$2$\left\{4,5\right\}$2$\left[1,2\right]$5$\left[0,2\right]$
3$\left\{4,5,6\right\}$3$\left\{4,5\right\}$3$\left[0,3\right]$6$\left[0,1\right]$