3.7.101.2. Flow models for ,

Table 3.7.23. Domains of the variables and minimum and maximum number of occurrences of each value for the constraint of FigureΒ 3.7.23.
$i$$\mathrm{\pi \pi \pi }\left({x}_{i}\right)$$i$$\mathrm{\pi \pi \pi }\left({x}_{i}\right)$$i$$\left[{\mathrm{\pi \pi \pi \pi }}_{i},{\mathrm{\pi \pi \pi \pi ₯}}_{i}\right]$$i$$\left[{\mathrm{\pi \pi \pi \pi }}_{i},{\mathrm{\pi \pi \pi \pi ₯}}_{i}\right]$
1$\left\{1,2\right\}$5$\left\{1,2,3\right\}$1$\left[1,2\right]$5$\left[0,2\right]$
2$\left\{1,2\right\}$6$\left\{2,3,4,5\right\}$2$\left[1,2\right]$
3$\left\{1,2\right\}$7$\left\{3,5\right\}$3$\left[1,1\right]$
4$\left\{1,2\right\}$4$\left[0,2\right]$
Table 3.7.23. Domains of the variables and minimum and maximum number of occurrences of each value for the constraint of FigureΒ 3.7.23.
$i$$\mathrm{\pi \pi \pi }\left({x}_{i}\right)$$i$$\mathrm{\pi \pi \pi }\left({x}_{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\}$5$\left\{1,2\right\}$loop$\left[2,2\right]$4$\left[1,2\right]$
2$\left\{1,2\right\}$6$\left\{2,4,5\right\}$1$\left[1,2\right]$5$\left[0,2\right]$
3$\left\{1,2\right\}$7$\left\{3,4,5\right\}$2$\left[2,3\right]$
4$\left\{1,2,3\right\}$3$\left[1,1\right]$

FigureΒ 3.7.23 presents flow models for the and the constraints. Blue arcs represent feasible flows respectively corresponding to the solutions $\left(\beta ©{x}_{1}=1,{x}_{2}=1,{x}_{3}=2,{x}_{4}=2,{x}_{5}=3,{x}_{6}=5,{x}_{7}=5\beta ͺ,\beta ©\mathrm{\pi \pi \pi }-1\mathrm{\pi \pi \pi \pi }-1\mathrm{\pi \pi \pi \pi ‘}-2,\mathrm{\pi \pi \pi }-2\mathrm{\pi \pi \pi \pi }-1\mathrm{\pi \pi \pi \pi ‘}-2,\mathrm{\pi \pi \pi }-3\mathrm{\pi \pi \pi \pi }-1\mathrm{\pi \pi \pi \pi ‘}-1,\mathrm{\pi \pi \pi }-4\mathrm{\pi \pi \pi \pi }-0\mathrm{\pi \pi \pi \pi ‘}-2,\mathrm{\pi \pi \pi }-5\mathrm{\pi \pi \pi \pi }-0\mathrm{\pi \pi \pi \pi ‘}-2\beta ͺ\right)$ and $\left(2,2,\beta ©{x}_{1}=1,{x}_{2}=2,{x}_{3}=2,{x}_{4}=2,{x}_{5}=1,{x}_{6}=4,{x}_{7}=3\beta ͺ,\beta ©\mathrm{\pi \pi \pi }-1\mathrm{\pi \pi \pi \pi }-1\mathrm{\pi \pi \pi \pi ‘}-2,\mathrm{\pi \pi \pi }-2\mathrm{\pi \pi \pi \pi }-2\mathrm{\pi \pi \pi \pi ‘}-3,\mathrm{\pi \pi \pi }-3\mathrm{\pi \pi \pi \pi }-1\mathrm{\pi \pi \pi \pi ‘}-1,\mathrm{\pi \pi \pi }-4\mathrm{\pi \pi \pi \pi }-1\mathrm{\pi \pi \pi \pi ‘}-2,\mathrm{\pi \pi \pi }-5\mathrm{\pi \pi \pi \pi }-0\mathrm{\pi \pi \pi \pi ‘}-2\beta ͺ\right)$, while pink arcs correspond to arcs that cannot carry any flow if the constraint has a solution:

• Within the context of the constraint variables ${x}_{1}$, ${x}_{2}$, ${x}_{3}$ and ${x}_{4}$ take their value within $\left\{1,2\right\}$. Since each value in $\left\{1,2\right\}$ can be used at most 2 times, variables different from ${x}_{1}$, ${x}_{2}$, ${x}_{3}$, ${x}_{4}$ cannot be assigned a value in $\left\{1,2\right\}$. Consequently, , , and . Since 3 is the only remaining value for ${x}_{3}$, and since value 3 should have no more than one occurrence, and are also forbidden.

• Note that, within the context of the we should have at least two assignments of the form ${x}_{i}=i$ ($i\beta \left[1,7\right]$). And ${x}_{1}$ and ${x}_{2}$ are the only two variables such that $i\beta \mathrm{\pi \pi \pi }\left({x}_{i}\right)$. Consequently and . Since we should have at least $1+2+1+1=5$ assignments of the form ${x}_{i}=j$ () and since only 5 variables can take a value in $\left[1,4\right]$, and .