3.7.101.4. Flow model for πšœπšŠπš–πšŽ_πšŠπš—πš_πšπš•πš˜πš‹πšŠπš•_πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’_πš•πš˜πš _πšžπš™

FigureΒ 3.7.25 presents a flow model for the πšœπšŠπš–πšŽ_πšŠπš—πš_πšπš•πš˜πš‹πšŠπš•_πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’_πš•πš˜πš _πšžπš™ constraint. Blue arcs represent a feasible flow corresponding to the solution πšœπšŠπš–πšŽ_πšŠπš—πš_πšπš•πš˜πš‹πšŠπš•_πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’_πš•πš˜πš _πšžπš™ (〈x 1 =2,x 2 =4,x 4 =βŒͺ,〈y 1 =2,y 2 =4,y 3 =4βŒͺ,βŒ©πšŸπšŠπš•-1 πš˜πš–πš’πš—-0 πš˜πš–πšŠπš‘-1,πšŸπšŠπš•-2 πš˜πš–πš’πš—-1 πš˜πš–πšŠπš‘-2,πšŸπšŠπš•-3 πš˜πš–πš’πš—-0 πš˜πš–πšŠπš‘-3,πšŸπšŠπš•-4 πš˜πš–πš’πš—-2 πš˜πš–πšŠπš‘-3,πšŸπšŠπš•-5 πš˜πš–πš’πš—-0 πš˜πš–πšŠπš‘-2,πšŸπšŠπš•-6 πš˜πš–πš’πš—-0 πš˜πš–πšŠπš‘-1βŒͺ), 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 1βˆ‰π‘‘π‘œπ‘š(y 1 )βˆͺπ‘‘π‘œπ‘š(y 2 )βˆͺπ‘‘π‘œπ‘š(y 3 ). Consequently x 1 =2 and, since y 1 is the only variable of {y 1 ,y 2 ,y 3 } that can be assigned value 2, the assignment y 1 =3 is forbidden. Now since 3βˆ‰π‘‘π‘œπ‘š(y 1 )βˆͺπ‘‘π‘œπ‘š(y 2 )βˆͺπ‘‘π‘œπ‘š(y 3 ) the assignment x 2 =3 is also forbidden. x 3 =6 is forbidden since 6βˆ‰π‘‘π‘œπ‘š(y 1 )βˆͺπ‘‘π‘œπ‘š(y 2 )βˆͺπ‘‘π‘œπ‘š(y 3 ). Finally x 3 =5 and y 3 =5 are also forbidden since value 4 must be assigned to at least two variables.

Figure 3.7.25. Flow model for the πšœπšŠπš–πšŽ_πšŠπš—πš_πšπš•πš˜πš‹πšŠπš•_πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’_πš•πš˜πš _πšžπš™ constraint described in tableΒ 3.7.25.
ctrs/flow_same_and_global_cardinality_low_up
Table 3.7.25. Domains of the variables and minimum and maximum number of occurrences of each value for the πšœπšŠπš–πšŽ_πšŠπš—πš_πšπš•πš˜πš‹πšŠπš•_πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’_πš•πš˜πš _πšžπš™ constraint of FigureΒ 3.7.25.
iπ‘‘π‘œπ‘š(x i )iπ‘‘π‘œπ‘š(y i )i[π‘œπ‘šπ‘–π‘› i ,π‘œπ‘šπ‘Žπ‘₯ i ]i[π‘œπ‘šπ‘–π‘› i ,π‘œπ‘šπ‘Žπ‘₯ i ]
1{1,2}1{2,3}1[0,1]4[2,3]
2{3,4}2{4,5}2[1,2]5[0,2]
3{4,5,6}3{4,5}3[0,3]6[0,1]