### 3.7.101.4. Flow model for $\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}$

Figure 3.7.25 presents a flow model for the $\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}$ constraint. Blue arcs represent a feasible flow corresponding to the solution $\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}$ $\left(〈{x}_{1}=2,{x}_{2}=4,{x}_{4}=〉,〈{y}_{1}=2,{y}_{2}=4,{y}_{3}=4〉,〈\mathrm{𝚟𝚊𝚕}-1\mathrm{𝚘𝚖𝚒𝚗}-0\mathrm{𝚘𝚖𝚊𝚡}-1,\mathrm{𝚟𝚊𝚕}-2\mathrm{𝚘𝚖𝚒𝚗}-1\mathrm{𝚘𝚖𝚊𝚡}-2,\mathrm{𝚟𝚊𝚕}-3\mathrm{𝚘𝚖𝚒𝚗}-0\mathrm{𝚘𝚖𝚊𝚡}-3,\mathrm{𝚟𝚊𝚕}-4\mathrm{𝚘𝚖𝚒𝚗}-2\mathrm{𝚘𝚖𝚊𝚡}-3,\mathrm{𝚟𝚊𝚕}-5\mathrm{𝚘𝚖𝚒𝚗}-0\mathrm{𝚘𝚖𝚊𝚡}-2,\mathrm{𝚟𝚊𝚕}-6\mathrm{𝚘𝚖𝚒𝚗}-0\mathrm{𝚘𝚖𝚊𝚡}-1〉\right)$, 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 $1\notin \mathrm{𝑑𝑜𝑚}\left({y}_{1}\right)\cup \mathrm{𝑑𝑜𝑚}\left({y}_{2}\right)\cup \mathrm{𝑑𝑜𝑚}\left({y}_{3}\right)$. 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 $3\notin \mathrm{𝑑𝑜𝑚}\left({y}_{1}\right)\cup \mathrm{𝑑𝑜𝑚}\left({y}_{2}\right)\cup \mathrm{𝑑𝑜𝑚}\left({y}_{3}\right)$ the assignment ${x}_{2}=3$ is also forbidden. ${x}_{3}=6$ is forbidden since $6\notin \mathrm{𝑑𝑜𝑚}\left({y}_{1}\right)\cup \mathrm{𝑑𝑜𝑚}\left({y}_{2}\right)\cup \mathrm{𝑑𝑜𝑚}\left({y}_{3}\right)$. 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 $\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}$ constraint of Figure 3.7.25.
$i$$\mathrm{𝑑𝑜𝑚}\left({x}_{i}\right)$$i$$\mathrm{𝑑𝑜𝑚}\left({y}_{i}\right)$$i$$\left[{\mathrm{𝑜𝑚𝑖𝑛}}_{i},{\mathrm{𝑜𝑚𝑎𝑥}}_{i}\right]$$i$$\left[{\mathrm{𝑜𝑚𝑖𝑛}}_{i},{\mathrm{𝑜𝑚𝑎𝑥}}_{i}\right]$
1$\left\{1,2\right\}$1$\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]$