### 3.7.101.2. Flow models for $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}$, $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}$

##### Table 3.7.23. Domains of the variables and minimum and maximum number of occurrences of each value for the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}$ constraint of Figure 3.7.23.
$i$$\mathrm{𝑑𝑜𝑚}\left({x}_{i}\right)$$i$$\mathrm{𝑑𝑜𝑚}\left({x}_{i}\right)$$i$$\left[{\mathrm{𝑜𝑚𝑖𝑛}}_{i},{\mathrm{𝑜𝑚𝑎𝑥}}_{i}\right]$$i$$\left[{\mathrm{𝑜𝑚𝑖𝑛}}_{i},{\mathrm{𝑜𝑚𝑎𝑥}}_{i}\right]$
1$\left\{1,2\right\}$5$\left\{1,2,3\right\}$1$\left[1,2\right]$5$\left[0,2\right]$
2$\left\{1,2\right\}$6$\left\{2,3,4,5\right\}$2$\left[1,2\right]$
3$\left\{1,2\right\}$7$\left\{3,5\right\}$3$\left[1,1\right]$
4$\left\{1,2\right\}$4$\left[0,2\right]$
##### Table 3.7.23. Domains of the variables and minimum and maximum number of occurrences of each value for the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}$ constraint of Figure 3.7.23.
$i$$\mathrm{𝑑𝑜𝑚}\left({x}_{i}\right)$$i$$\mathrm{𝑑𝑜𝑚}\left({x}_{i}\right)$$i$$\left[{\mathrm{𝑜𝑚𝑖𝑛}}_{i},{\mathrm{𝑜𝑚𝑎𝑥}}_{i}\right]$$i$$\left[{\mathrm{𝑜𝑚𝑖𝑛}}_{i},{\mathrm{𝑜𝑚𝑎𝑥}}_{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]$

Figure 3.7.23 presents flow models for the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}$ and the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}$ constraints. Blue arcs represent feasible flows respectively corresponding to the solutions $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}$ $\left(〈{x}_{1}=1,{x}_{2}=1,{x}_{3}=2,{x}_{4}=2,{x}_{5}=3,{x}_{6}=5,{x}_{7}=5〉,〈\mathrm{𝚟𝚊𝚕}-1\mathrm{𝚘𝚖𝚒𝚗}-1\mathrm{𝚘𝚖𝚊𝚡}-2,\mathrm{𝚟𝚊𝚕}-2\mathrm{𝚘𝚖𝚒𝚗}-1\mathrm{𝚘𝚖𝚊𝚡}-2,\mathrm{𝚟𝚊𝚕}-3\mathrm{𝚘𝚖𝚒𝚗}-1\mathrm{𝚘𝚖𝚊𝚡}-1,\mathrm{𝚟𝚊𝚕}-4\mathrm{𝚘𝚖𝚒𝚗}-0\mathrm{𝚘𝚖𝚊𝚡}-2,\mathrm{𝚟𝚊𝚕}-5\mathrm{𝚘𝚖𝚒𝚗}-0\mathrm{𝚘𝚖𝚊𝚡}-2〉\right)$ and $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}$ $\left(2,2,〈{x}_{1}=1,{x}_{2}=2,{x}_{3}=2,{x}_{4}=2,{x}_{5}=1,{x}_{6}=4,{x}_{7}=3〉,〈\mathrm{𝚟𝚊𝚕}-1\mathrm{𝚘𝚖𝚒𝚗}-1\mathrm{𝚘𝚖𝚊𝚡}-2,\mathrm{𝚟𝚊𝚕}-2\mathrm{𝚘𝚖𝚒𝚗}-2\mathrm{𝚘𝚖𝚊𝚡}-3,\mathrm{𝚟𝚊𝚕}-3\mathrm{𝚘𝚖𝚒𝚗}-1\mathrm{𝚘𝚖𝚊𝚡}-1,\mathrm{𝚟𝚊𝚕}-4\mathrm{𝚘𝚖𝚒𝚗}-1\mathrm{𝚘𝚖𝚊𝚡}-2,\mathrm{𝚟𝚊𝚕}-5\mathrm{𝚘𝚖𝚒𝚗}-0\mathrm{𝚘𝚖𝚊𝚡}-2〉\right)$, while pink arcs correspond to arcs that cannot carry any flow if the constraint has a solution:

• Within the context of the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}$ 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, ${x}_{3}\ne 1$, ${x}_{3}\ne 2$, ${x}_{4}\ne 1$ and ${x}_{4}\ne 2$. Since 3 is the only remaining value for ${x}_{3}$, and since value 3 should have no more than one occurrence, ${x}_{4}\ne 3$ and ${x}_{5}\ne 3$ are also forbidden.

• Note that, within the context of the $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}$ we should have at least two assignments of the form ${x}_{i}=i$ ($i\in \left[1,7\right]$). And ${x}_{1}$ and ${x}_{2}$ are the only two variables such that $i\in \mathrm{𝑑𝑜𝑚}\left({x}_{i}\right)$. Consequently ${x}_{1}\ne 2$ and ${x}_{2}\ne 1$. Since we should have at least $1+2+1+1=5$ assignments of the form ${x}_{i}=j$ ($i\ne j,j\in \left[1,4\right]$) and since only 5 variables can take a value in $\left[1,4\right]$, ${x}_{6}\ne 4$ and ${x}_{7}\ne 5$.