உੑ i ͱঁੑ j ͕Ϛονϯά x ʹ͓͚Δ ϒϩοΩϯάϖΞ ⇔ (1) i ͱ j ݱࡏϚον͍ͯ͠ͳ͍ (2) i ಠ or ݱࡏͷ૬खΑΓ j ΛΉ (3) j ಠ or ݱࡏͷ૬खΑΓ i ΛΉ 1 2 3 A B C 1 2 2 1 3 2 1 B A C A B C B A ଟରҰϚονϯάʢஉੑΛֶੜɼঁੑΛύʔτʹஔ͖͑ʣ ʹ͓͚ΔϒϩοΩϯάϖΞ... (1), (2) (3’) ύʔτ j ʹଐ͞Ε͍ͯΔֶੜͷਓఆһͷ্ݶະຬ or ݱࡏଐ͞Ε͍ͯΔֶੜΑΓֶੜ i ΛΉ 6 / 15
உੑ i ͱঁੑ j ͕Ϛονϯά x ʹ͓͚Δ ϒϩοΩϯάϖΞ ⇔ (1) i ͱ j ݱࡏϚον͍ͯ͠ͳ͍ (2) i ಠ or ݱࡏͷ૬खΑΓ j ΛΉ (3) j ಠ or ݱࡏͷ૬खΑΓ i ΛΉ 1 2 3 A B C 1 2 2 1 3 2 1 B A C A B C B A ଟରҰϚονϯάʢஉੑΛֶੜɼঁੑΛύʔτʹஔ͖͑ʣ ʹ͓͚ΔϒϩοΩϯάϖΞ... (1), (2) (3’) ύʔτ j ʹଐ͞Ε͍ͯΔֶੜͷਓఆһͷ্ݶະຬ or ݱࡏଐ͞Ε͍ͯΔֶੜΑΓֶੜ i ΛΉ 6 / 15
உੑ i ͱঁੑ j ͕Ϛονϯά x ʹ͓͚Δ ϒϩοΩϯάϖΞ ⇔ (1) i ͱ j ݱࡏϚον͍ͯ͠ͳ͍ (2) i ಠ or ݱࡏͷ૬खΑΓ j ΛΉ (3) j ಠ or ݱࡏͷ૬खΑΓ i ΛΉ 1 2 3 A B C 1 2 2 1 3 2 1 B A C A B C B A ଟରҰϚονϯάʢஉੑΛֶੜɼঁੑΛύʔτʹஔ͖͑ʣ ʹ͓͚ΔϒϩοΩϯάϖΞ... (1), (2) (3’) ύʔτ j ʹଐ͞Ε͍ͯΔֶੜͷਓఆһͷ্ݶະຬ or ݱࡏଐ͞Ε͍ͯΔֶੜΑΓֶੜ i ΛΉ 6 / 15
உੑ i ͱঁੑ j ͕Ϛονϯά x ʹ͓͚Δ ϒϩοΩϯάϖΞ ⇔ (1) i ͱ j ݱࡏϚον͍ͯ͠ͳ͍ (2) i ಠ or ݱࡏͷ૬खΑΓ j ΛΉ (3) j ಠ or ݱࡏͷ૬खΑΓ i ΛΉ 1 2 3 A B C 1 2 2 1 3 2 1 B A C A B C B A ۦ͚མͪ ଟରҰϚονϯάʢஉੑΛֶੜɼঁੑΛύʔτʹஔ͖͑ʣ ʹ͓͚ΔϒϩοΩϯάϖΞ... (1), (2) (3’) ύʔτ j ʹଐ͞Ε͍ͯΔֶੜͷਓఆһͷ্ݶະຬ or ݱࡏଐ͞Ε͍ͯΔֶੜΑΓֶੜ i ΛΉ 6 / 15
உੑ i ͱঁੑ j ͕Ϛονϯά x ʹ͓͚Δ ϒϩοΩϯάϖΞ ⇔ (1) i ͱ j ݱࡏϚον͍ͯ͠ͳ͍ (2) i ಠ or ݱࡏͷ૬खΑΓ j ΛΉ (3) j ಠ or ݱࡏͷ૬खΑΓ i ΛΉ 1 2 3 A B C 1 2 2 1 3 2 1 B A C A B C B A ଟରҰϚονϯάʢஉੑΛֶੜɼঁੑΛύʔτʹஔ͖͑ʣ ʹ͓͚ΔϒϩοΩϯάϖΞ... (1), (2) (3’) ύʔτ j ʹଐ͞Ε͍ͯΔֶੜͷਓఆһͷ্ݶະຬ or ݱࡏଐ͞Ε͍ͯΔֶੜΑΓֶੜ i ΛΉ 6 / 15
Tuba Cb Perc set of parts 0 2 4 6 8 10 12 # of students minimum quota maximum quota (a,b)=(0.5,0.5) → Fl Ob Fg Cl Sax Tp Hr Tb Euph Tuba Cb Perc set of parts 0 2 4 6 8 10 12 # of students minimum quota maximum quota (a,b,c)=(0.1,0.5,0.4) • ܭࢉ࣌ؒ: 0.05 ඵ • ϒϩοΩϯάϖΞ 6 • શͯͷֶੜɼୈ 3 رҎ্ͷύʔτʹׂΓͯΒΕͨ • ઌͷ݁ՌͱൺΔͱɼฤͷόϥϯεվળ 11 / 15
and M. Balinski, “The stable admissions polytope,” Mathematical Programming, Ser.A, Vol.87, pp.427–439, 2000. • M. Delorme et al., “Mathematical models for stable matching problems with ties and incomplete lists,” European Journal of Operational Research, Vol.277, pp.426–441, 2019. 13 / 15