https://youtu.be/1ZRsualHTxY

1. ήʔϜཧ࿦#4*$ୈճ
ଓɾΧʔωϧ

2. ఏ‫ܞ‬ͷෆຬͱϓϨΠϠʔͷ༏Ґੑ ෮श
Χʔωϧͷఆٛ ෮श
ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

3. ఏ‫ܞ‬ͷෆຬͱϓϨΠϠʔͷ༏Ґੑ
ఆٛɿఏ‫ܞ‬

ͷෆຬ
ಛੑؔ਺‫ܗ‬ήʔϜ

ʹ͓͍ͯ
഑෼

ͱఏ‫ܞ‬

ΛͱΔͱ͖

Λ഑෼

ʹର͢Δఏ‫ܞ‬

ͷෆຬͱ͍͏
ਓͷϓϨΠϠʔ


ΛͱΓ

ͱ͓͘
ఆٛɿ࠷େෆຬ
ਓͷϓϨΠϠʔ


ͱ഑෼

ʹ͍ͭͯ

Λ഑෼

ʹ͓͚ΔϓϨΠϠʔ

ͷ

ʹର͢Δ࠷େෆຬͱ͍͏
S
(N, v) x ∈
𝒜
(v) S ⊆ N
e(S, x) = v(S) −
∑
i∈S
xi x S
i j ∈ N Tij = {S ⊆ N|i ∈ S, j ∉ S}
i j ∈ N x ∈
𝒜
(v)
sij(x) = max
S∈Tij
e(S, x)
x i j

4. ఏ‫ܞ‬ͷෆຬͱϓϨΠϠʔͷ༏Ґੑ
ఆٛɿ༏Ґ
ਓͷϓϨΠϠʔ


ͱ഑෼

ʹ͍ͭͯ


ͱͳΔͱ͖
഑෼

ʹ͓͍ͯϓϨΠϠʔ

͸

ΑΓ΋༏ҐͰ͋Δͱ͍͍

ͱॻ͘

Ͱ͋Ε͹
ϓϨΠϠʔ

͸

ʹରͯ͠རಘΛ͘Εͱ͸͍͑ͳ͍
ͳͥͳΒ

͸ఏ‫ܞ‬Λ૊ΉΠϯηϯςΟϒ͕ͳ͘ͳΔͨΊ
i j ∈ N x ∈
𝒜
(v)
sij(x) sji(x) xj v({j})
x i j i ≻x j
xj ≤ v({j}) i j
j
ਓͷϓϨΠϠʔͷ
࠷େෆຬΛൺֱ

5. Χʔωϧͷఆٛ
ఆٛɿ‫ߧۉ‬ঢ়ଶ
ਓͷϓϨΠϠʔ


ͱ഑෼

ʹ͍ͭͯ

Ͱ΋ͳ͘

Ͱ΋ͳ͍ͱ͖
഑෼

ʹ͓͍ͯϓϨΠϠʔ

ͱ

͸‫ߧۉ‬ঢ়ଶͰ͋Δͱ͍͍

ͱॻ͘
ఆٛɿΧʔωϧ
೚ҙͷਓͷϓϨΠϠʔ͕‫ߧۉ‬ঢ়ଶʹ͋ΔΑ͏ͳ഑෼ͷશମ

ΛΧʔωϧͱ͍͏
i j ∈ N x ∈
𝒜
(v)
i ≻x j j ≻x i
x i j i ∼x j
𝒦
= {x ∈
𝒜
(v)|i ∼x j ∀i, j ∈ N, i ≠ j}

6. Χʔωϧͷఆٛ
ఆٛɿ‫ߧۉ‬ঢ়ଶ
ਓͷϓϨΠϠʔ


ͱ഑෼

ʹ͍ͭͯ

Ͱ΋ͳ͘

Ͱ΋ͳ͍ͱ͖
഑෼

ʹ͓͍ͯϓϨΠϠʔ

ͱ

͸‫ߧۉ‬ঢ়ଶͰ͋Δͱ͍͍

ͱॻ͘

ͱͳΔͨΊʹ͸

͸഑෼ͷ‫ݸ‬ਓ߹ཧੑΑΓ


ʹ஫ҙͯ͠

͔ͭ

͢ͳΘͪ

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i j ∈ N x ∈
𝒜
(v)
i ≻x j j ≻x i
x i j i ∼x j
i ∼x j x ∈
𝒜
(v) xi ≥ v({i}) xj ≥ v({j})
sij(x) ≥ sji(x) sij(x) ≤ sji(x) sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
ఆٛɿ༏Ґ
ਓͷϓϨΠϠʔ


ͱ഑෼

ʹ͍ͭͯ


ͱͳΔͱ͖
഑෼

ʹ͓͍ͯϓϨΠϠʔ

͸

ΑΓ΋༏ҐͰ͋Δͱ͍͍

ͱॻ͘
i j ∈ N x ∈
𝒜
(v)
sij(x) sji(x) xj v({j})
x i j i ≻x j

͔ͭ

xj ≥ v({j}) xj ≤ v({j})

͔ͭ

xi ≥ v({i}) xi ≤ v({i})

7. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}
1
2 3
ίΞ
x1 = 10
x2 = 3
x3 = 0
ަবू߹

8. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}

ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸





x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) = max(v({1,3}) − x1 − x3), v({1}) − x1) )
= max(7 − x1 − x3, 0 − x1 )
=
{
7 − x1 − x3 (0 ≤ x3 ≤ 7)
−x1 (7 x3 ≤ 10)
s13(x) = max(v({1,2}) − x1 − x2, v({1}) − x1 )
= 10 − x1 − x2

9. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}

ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸







x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) =
{
7 − x1 − x3 (0 ≤ x3 ≤ 7)
−x1 (7 x3 ≤ 10)
s13(x) = 10 − x1 − x2
s21(x) = max(v({2,3}) − x2 − x3, v({2}) − x2 )
= max(0 − x2 − x3, 0 − x2 )
= − x2
s23(x) = max(v({1,2}) − x1 − x2, v({2}) − x2 )
= 10 − x1 − x2

10. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}

ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) =
{
7 − x1 − x3 (0 ≤ x3 ≤ 7)
−x1 (7 x3 ≤ 10)
s13(x) = 10 − x1 − x2
s21(x) = − x2
s23(x) = 10 − x1 − x2
s31(x) = − x3
s32(x) =
{
7 − x1 − x3 (0 ≤ x1 ≤ 7)
−x3 (7 x1 ≤ 10)

11. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}

ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) =
{
7 − x1 − x3 (0 ≤ x3 ≤ 7)
−x1 (7 x3 ≤ 10)
s13(x) = 10 − x1 − x2
s21(x) = − x2
s23(x) = 10 − x1 − x2
s31(x) = − x3
s32(x) =
{
7 − x1 − x3 (0 ≤ x1 ≤ 7)
−x3 (7 x1 ≤ 10)

ͱͳΔͨΊʹ͸

ͷͱ͖
ɾ

ͱͳΓෆద
ɾ

ͱͳΓ
ɹ

͔ͭ

ͷͱ͖

Ͱ͋ΔͨΊ

ͱͳΒͳ͍
ɾ

1 ∼x 2
0 ≤ x3 ≤ 7
s12(x) s21(x) ⇔ 7 − x1 − x3 − x2 ⇔ x2
3
2
0 = v({2})
s12(x) s21(x) ⇔ 7 − x1 − x3 − x2 ⇔ x2
3
2
0 ≤ x3 ≤ 7 x2
3
2
3
2
x1 x1 = v({1}) = 0
s12(x) = s21(x) ⇔ 7 − x1 − x3 = − x2 ⇔ x2 =
3
2

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
഑෼ͷશମ߹ཧੑɿ

x1 + x2 + x3 = 10

12. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}

ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) =
{
7 − x1 − x3 (0 ≤ x3 ≤ 7)
−x1 (7 x3 ≤ 10)
s13(x) = 10 − x1 − x2
s21(x) = − x2
s23(x) = 10 − x1 − x2
s31(x) = − x3
s32(x) =
{
7 − x1 − x3 (0 ≤ x1 ≤ 7)
−x3 (7 x1 ≤ 10)

ͱͳΔͨΊʹ͸

ͷͱ͖


ͷͱ͖
ɾ

ͱͳΓෆద
ɾ

ͳΒ͹

͕ͩ
ɹ

ͱͳΔͨΊෆద
ɾ

1 ∼x 2
0 ≤ x3 ≤ 7 x2 =
3
2
7 x3 ≤ 10
s12(x) s21(x) ⇔ − x1 − x2 ⇔ x2 x1 ≥ 0 = v({1})
s12(x) s21(x) ⇔ − x1 − x2 ⇔ x2 x1 x1 = v({1}) = 0
x2 0
s12(x) = s21(x) ⇔ − x1 = − x2 ⇔ x2 = x1

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})

13. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}

ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) =
{
7 − x1 − x3 (0 ≤ x3 ≤ 7)
−x1 (7 x3 ≤ 10)
s13(x) = 10 − x1 − x2
s21(x) = − x2
s23(x) = 10 − x1 − x2
s31(x) = − x3
s32(x) =
{
7 − x1 − x3 (0 ≤ x1 ≤ 7)
−x3 (7 x1 ≤ 10)

ͱͳΔͨΊʹ͸

ͷͱ͖


ͷͱ͖

1 ∼x 2
0 ≤ x3 ≤ 7 x2 =
3
2
7 x3 ≤ 10 x2 = x1

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})

14. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}

ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) =
{
7 − x1 − x3 (0 ≤ x3 ≤ 7)
−x1 (7 x3 ≤ 10)
s13(x) = 10 − x1 − x2
s21(x) = − x2
s23(x) = 10 − x1 − x2
s31(x) = − x3
s32(x) =
{
7 − x1 − x3 (0 ≤ x1 ≤ 7)
−x3 (7 x1 ≤ 10)

ͱͳΔͨΊʹ͸ʜ

ͷͱ͖


ͷͱ͖


ͱͳΔͨΊʹ͸
ɾ

ͱͳΓෆద
ɾ

ͱͳΓෆద
ɾ

1 ∼x 2 0 ≤ x3 ≤ 7 x2 =
3
2
7 x3 ≤ 10 x2 = x1
1 ∼x 3
s13(x) s31(x) ⇔ 10 − x1 − x2 − x3 ⇔ x3 0 = v({3})
s13(x) s31(x) ⇔ 10 − x1 − x2 − x3 ⇔ x3 0 = v({3})
s13(x) = s31(x) ⇔ 10 − x1 − x2 = − x3 ⇔ x3 = 0

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})

15. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}

ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) =
{
7 − x1 − x3 (0 ≤ x3 ≤ 7)
−x1 (7 x3 ≤ 10)
s13(x) = 10 − x1 − x2
s21(x) = − x2
s23(x) = 10 − x1 − x2
s31(x) = − x3
s32(x) =
{
7 − x1 − x3 (0 ≤ x1 ≤ 7)
−x3 (7 x1 ≤ 10)

ͱͳΔͨΊʹ͸ʜ

ͷͱ͖


ͷͱ͖


ͱͳΔͨΊʹ͸

1 ∼x 2 0 ≤ x3 ≤ 7 x2 =
3
2
7 x3 ≤ 10 x2 = x1
1 ∼x 3
x3 = 0

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})

16. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}

ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) =
{
7 − x1 − x3 (0 ≤ x3 ≤ 7)
−x1 (7 x3 ≤ 10)
s13(x) = 10 − x1 − x2
s21(x) = − x2
s23(x) = 10 − x1 − x2
s31(x) = − x3
s32(x) =
{
7 − x1 − x3 (0 ≤ x1 ≤ 7)
−x3 (7 x1 ≤ 10)

ͱͳΔͨΊʹ͸ʜ

ͷͱ͖


ͷͱ͖


ͱͳΔͨΊʹ͸ʜ


ͱͳΔͨΊʹ͸

ͷͱ͖
ɾ

ͱͳΓ

ͱ͢Δͱ

ͱͳΓ

ͱ߹ΘͤΔͱ

Ͱͳ͚Ε͹ͳΒͣෆద
ɾ

ͱͳΓ

ͱ͢Δͱ

ͱͳΓෆద
ɾ

1 ∼x 2 0 ≤ x3 ≤ 7 x2 =
3
2
7 x3 ≤ 10 x2 = x1
1 ∼x 3 x3 = 0
2 ∼x 3
0 ≤ x1 ≤ 7
s23(x) s32(x) ⇔ 10 − x1 − x2 7 − x1 − x3 ⇔ 3 + x3 x2 x3 = v({3}) = 0
3 x2 0 ≤ x1 ≤ 7 x3 0
s23(x) s32(x) ⇔ 10 − x1 − x2 7 − x1 − x3 ⇔ 3 + x3 x2 x2 = v({2}) = 0
3 + x3 0 ⇔ x3 − 3 0
s23(x) = s32(x) ⇔ 10 − x1 − x2 = 7 − x1 − x3 ⇔ 3 + x3 = x2

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})

17. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}

ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) =
{
7 − x1 − x3 (0 ≤ x3 ≤ 7)
−x1 (7 x3 ≤ 10)
s13(x) = 10 − x1 − x2
s21(x) = − x2
s23(x) = 10 − x1 − x2
s31(x) = − x3
s32(x) =
{
7 − x1 − x3 (0 ≤ x1 ≤ 7)
−x3 (7 x1 ≤ 10)

ͱͳΔͨΊʹ͸ʜ

ͷͱ͖


ͷͱ͖


ͱͳΔͨΊʹ͸ʜ


ͱͳΔͨΊʹ͸ʜ

ͷͱ͖


ͷͱ͖
ɾ

ͱͳΓ

ͱͳΒͣෆద
ɾ

ͱͳΓ
ෆద
ɾ

1 ∼x 2 0 ≤ x3 ≤ 7 x2 =
3
2
7 x3 ≤ 10 x2 = x1
1 ∼x 3 x3 = 0
2 ∼x 3 0 ≤ x1 ≤ 7 3 + x3 = x2
7 x1 ≤ 10
s23(x) s32(x) ⇔ 10 − x1 − x2 − x3 ⇔ x3 0 x3 = v({3}) = 0
s23(x) s32(x) ⇔ 10 − x1 − x2 − x3 ⇔ x3 0
s23(x) = s32(x) ⇔ 10 − x1 − x2 = − x3 ⇔ x3 = 0

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})

18. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}

ͱͳΔͨΊʹ͸ʜ

ͷͱ͖


ͷͱ͖


ͱͳΔͨΊʹ͸ʜ


ͱͳΔͨΊʹ͸ʜ

ͷͱ͖


ͷͱ͖


͸഑෼͔ͩΒ

ΑΓ

Ҏ্ΑΓΧʔωϧ

1 ∼x 2 0 ≤ x3 ≤ 7 x2 =
3
2
7 x3 ≤ 10 x2 = x1
1 ∼x 3 x3 = 0
2 ∼x 3 0 ≤ x1 ≤ 7 3 + x3 = x2 7 x1 ≤ 10 x3 = 0
x = (x1, x2, x3) ∈
𝒜
(v) x1 + x2 + x3 = 10 x1 = 10 −
3
2
=
17
2
𝒦
(v) = {x ∈
𝒜
(v)|(x1, x2, x3) = (
17
2
,
3
2
,0)}

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
1
2 3
ίΞ
x1 = 10
x2 = 3
x3 = 0
ަবू߹
Χʔωϧ

19. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = v({2,3}) = 10, v({1}) = v({2}) = v({3}) = 0
C(v) = ∅
1
2 3
x1 = 0
x2 = 0
x3 = 0

(x1, x2, x3) =
(
10
3
,
10
3
,
10
3 )
ަবू߹

20. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = v({2,3}) = 10, v({1}) = v({2}) = v({3}) = 0
C(v) = ∅

ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸








x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) = max(v({1,3}) − x1 − x3), v({1}) − x1) )
= max(10 − x1 − x3, 0 − x1 )
= 10 − x1 − x3
s13(x) = 10 − x1 − x2
s21(x) = 10 − x2 − x3
s23(x) = 10 − x1 − x2
s31(x) = 10 − x2 − x3
s32(x) = 10 − x1 − x3

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})

21. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = v({2,3}) = 10, v({1}) = v({2}) = v({3}) = 0
C(v) = ∅

ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) = 10 − x1 − x3
s13(x) = 10 − x1 − x2
s21(x) = 10 − x2 − x3
s23(x) = 10 − x1 − x2
s31(x) = 10 − x2 − x3
s32(x) = 10 − x1 − x3

ͱͳΔͨΊʹ͸
ɾ


Ͱͳ͚Ε͹ͳΒͳ͍͕

ͱͳΓෆద
ɾ


Ͱͳ͚Ε͹ͳΒͳ͍͕

ͱͳΓෆద
ɾ

1 ∼x 2
s12(x) s21(x) ⇔ 10 − x1 − x3 10 − x2 − x3 ⇔ x2 x1
x2 = v({2}) = 0 x2 = 0 x1
s12(x) s21(x) ⇔ 10 − x1 − x3 10 − x2 − x3 ⇔ x2 x1
x1 = v({1}) = 0 x1 = 0 x2
s12(x) = s21(x) ⇔ 10 − x1 − x3 = 10 − x2 − x3 ⇔ x2 = x1

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})

22. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = v({2,3}) = 10, v({1}) = v({2}) = v({3}) = 0
C(v) = ∅

ͱͳΔͨΊʹ͸ʜ


ͱͳΔͨΊʹ͸ʜ


ͱͳΔͨΊʹ͸ʜ


͸഑෼͔ͩΒ

ΑΓ

Ҏ্ΑΓΧʔωϧ

1 ∼x 2 x2 = x1
1 ∼x 3 x3 = x1
2 ∼x 3 x2 = x3
x = (x1, x2, x3) ∈
𝒜
(v) x1 + x2 + x3 = 10 x1 = x2 = x3 =
10
3
𝒦
(v) = {x ∈
𝒜
(v)|(x1, x2, x3) = (
10
3
,
10
3
,
10
3
)}

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
1
2 3
x1 = 0
x2 = 0
x3 = 0

(x1, x2, x3) =
(
10
3
,
10
3
,
10
3 )
ަবू߹͔ͭΧʔωϧ

23. ήʔϜཧ࿦#4*$ୈճ
ଓɾΧʔωϧ
࣍ճɿަবू߹ͱΧʔωϧ