1 [1, 2, 3, 4, 5, 8, 9, 10, 12, 15] The Boston Public Schools system, BPS (Deferred Acceptance system, DA) (Top Trading Cycles system, TTC) cf. [13] [
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1 Vol.2, No.x, April 2015, pp.xx-xx ISSN xxxx-xxxx Tel ( ) Fax
2 1 [1, 2, 3, 4, 5, 8, 9, 10, 12, 15] The Boston Public Schools system, BPS (Deferred Acceptance system, DA) (Top Trading Cycles system, TTC) cf. [13] [1] cf. [7, 14, 15, 11] cf. [13, 7] 3 BPS DA TTC cf. [13] TTC DA cf. [14] 3 DA TTC DA DA (DA) cf. [13] DA cf. [13] DA (cf. [13]) 2
3 3 4 BPS, DA, TTC 3 cf. [2] GPA GPA [9] 1 1 cf. [13] GPA 3
4 TTC cf. [2] cf. [2] DA n = 204 m = 9 i j x ij = 1 x ij = 0 {0, 1}- b = 25 i j c ij max. s.t. n m c ij x ij (2.1) i=1 j=1 n x ij b j (j = 1,, m) (2.2) i=1 m x ij = 1 (i = 1,, n) (2.3) j=1 x ij {0, 1} (i = 1,, n, j = 1,, m) (2.4) (2.4) {0, 1}- (2.2) (2.3) (2.1) (2.2) (2.4) max. m c ij x ij ( i {1, 2,, n}) (2.5) j=1 1 (2.1) (2.5) cf. [6] 4
5 [8, 9] cf. [2] [ 1 2 ] = 40 > 30 = [ 2 3 ] cf. [2, 3] 2.1: 10 \
6 Excel python2.7 gurobi bit CPU Intel(R) Core(TM) i7 [2.93GHz] 4GB : d1 d2 d3 d4 d5 d6 d7 d8 d9 d ,
7 2.3: d1 d2 d3 d4 d5 d6 d7 d8 d9 d : d1 d2 d3 d4 d5 d6 d7 d8 d9 d
8 2.3 1 GPA GPA 2 1 N(2, 1) 10 GPA GPA , 0.68) 3 (1.88, (1.87, GPA : GPA 0.2 GPA GPA GPA [9] 2 A, B GPA GPA A B
9 (2.1) (A 1, B 2 ) = (B 1, A 2 ) A B (A 1, B 2 ) > (B 1, A 2 ) A GP A, 1.5 GP A, 1 GP A A B (A 1, B 2 ) > (B 1, A 2 ) A , GPA 10 GPA (2) 1(1) GPA 4 9(2) 9 2 GPA 1(1) % 8% 9
10 2.5: 1 GPA 4 9(2) 1(1) 9 6(2) 1(1) 11 5(2) 1(1) 5 9(2) 3(1) 1 7(3) 1(1) 23 5(2) 1(1) 11 6(2) 5(1) 25 9(3) 1(1) 12 7(2) 1(1) 6 4(2) 3(1) 40 9(2) 3(1) 16 7(3) 5(1) 40 6(2) 3(1) 22 7(2) 1(1) 7 6(2) 3(1) 52 9(2) 3(1) 50 9(3) 6(2) 95 1(1) 8(3) 31 7(2) 3(1) 21 7(2) 2(1) 57 5(2) 6(1) 59 5(1) 6(2) 106 9(3) 3(1) 47 8(2) 3(1) 29 5(2) 3(1) 62 9(2) 3(1) 86 9(3) 1(1) 127 5(2) 1(1) 53 8(2) 5(1) 46 4(2) 1(1) 77 5(2) 2(1) 111 9(2) 4(1) 142 3(1) 9(3) 92 6(2) 2(1) 48 7(3) 1(1) 79 4(2) 2(1) 147 2(2) 3(1) 148 1(1) 5(2) 113 2(1) 7(2) 63 3(1) 6(2) 95 3(1) 8(2) 149 8(3) 5(1) 149 6(2) 1(1) 117 3(1) 7(2) 86 2(1) 9(2) 151 2(1) 5(2) 154 4(1) 7(2) 189 1(1) 6(2) 120 1(1) 4(2) 125 1(1) 6(2) 153 1(1) 4(2) 159 1(1) 7(3) 198 1(1) 6(2) 156 3(1) 4(2) 133 3(1) 5(2) 154 3(1) 5(2) 160 1(1) 2(2) 201 3(1) 5(2) 157 5(1) 4(2) 144 1(1) 9(3) 155 2(1) 8(2) 182 5(1) 8(3) 158 3(1) 4(2) 157 6(2) 8(3) 165 3(1) 9(2) 189 5(1) 7(3) 187 6(2) 3(1) 168 3(1) 4(2) 183 1(1) 5(2) 196 3(1) 8(3) 196 3(1) 6(2) 196 1(1) 4(2) 197 6(1) 8(2) 203 1(1) 6(2) 7 7(3) 2(1) 4 7(2) 2(1) 4 8(3) 3(1) 13 8(3) 2(1) 1 6(2) 1(1) 15 4(2) 2(1) 6 8(2) 3(1) 12 6(2) 3(1) 14 8(3) 3(1) 13 9(2) 2(1) 16 7(2) 3(1) 22 5(2) 3(1) 19 7(2) 1(1) 50 9(2) 1(1) 50 8(2) 5(1) 27 7(2) 5(1) 72 9(2) 1(1) 34 6(2) 3(1) 62 8(2) 4(1) 58 9(2) 4(1) 42 9(3) 2(1) 77 9(2) 1(1) 46 9(2) 1(1) 66 1(1) 7(2) 87 9(2) 2(1) 45 7(2) 5(1) 88 9(2) 1(1) 64 5(2) 2(1) 71 8(2) 4(1) 92 5(1) 8(2) 62 5(1) 7(2) 113 1(1) 9(2) 86 3(1) 5(2) 72 5(2) 3(1) 95 4(1) 8(2) 65 2(1) 8(3) 127 2(1) 9(2) 123 1(1) 9(2) 84 9(2) 4(1) 132 1(1) 6(2) 86 9(3) 1(1) 130 1(1) 9(2) 128 4(2) 2(1) 112 4(1) 8(2) 159 2(1) 7(2) 118 4(2) 1(1) 132 1(1) 9(2) 141 3(1) 6(2) 113 4(1) 7(2) 182 2(1) 7(2) 139 1(1) 4(2) 143 1(1) 9(2) 150 2(1) 9(3) 125 5(2) 2(1) 150 2(1) 4(2) 158 6(2) 1(1) 151 3(1) 4(2) 128 2(1) 5(2) 169 3(1) 7(2) 172 3(1) 5(2) 171 1(1) 9(2) 144 4(1) 9(2) 175 2(1) 8(3) 202 3(1) 6(2) 197 2(1) 6(2) 167 3(1) 7(3) 195 1(1) 8(3) 186 3(1) 5(2) 202 5(1) 8(2) 197 2(1) 9(3) d3, d6, d
11 2.6: 1 3 GPA 4 9(2) 1(1) 9 6(2) 1(1) 11 5(2) 1(1) 5 9(2) 3(1) 1 7(3) 1(1) 23 5(2) 1(1) 11 6(2) 5(1) 25 9(3) 1(1) 12 7(2) 1(1) 6 4(2) 3(1) 40 9(2) 3(1) 16 7(3) 5(1) 40 6(2) 3(1) 22 7(2) 1(1) 7 6(2) 3(1) 52 9(2) 3(1) 50 9(3) 6(2) 68 9(3) 1(1) 31 7(2) 3(1) 21 7(2) 2(1) 57 5(2) 6(1) 59 5(1) 6(2) 95 1(1) 8(3) 47 8(2) 3(1) 29 5(2) 3(1) 62 9(2) 3(1) 86 9(3) 1(1) 106 9(3) 3(1) 53 8(2) 5(1) 46 4(2) 1(1) 77 5(2) 2(1) 111 9(2) 4(1) 139 3(1) 9(3) 92 6(2) 2(1) 48 7(3) 1(1) 79 4(2) 2(1) 147 2(2) 3(1) 142 3(1) 9(3) 113 2(1) 7(2) 63 3(1) 6(2) 95 3(1) 8(2) 149 8(3) 5(1) 148 1(1) 5(2) 117 3(1) 7(2) 86 2(1) 9(2) 151 2(1) 5(2) 154 4(1) 7(2) 149 6(2) 1(1) 120 1(1) 4(2) 125 1(1) 6(2) 153 1(1) 4(2) 159 1(1) 7(3) 189 1(1) 6(2) 156 3(1) 4(2) 133 3(1) 5(2) 154 3(1) 5(2) 160 1(1) 2(2) 198 1(1) 6(2) 157 5(1) 4(2) 144 1(1) 9(3) 155 2(1) 8(2) 182 5(1) 8(3) 158 3(1) 4(2) 157 6(2) 8(3) 165 3(1) 9(2) 189 5(1) 7(3) 187 6(2) 3(1) 168 3(1) 4(2) 183 1(1) 5(2) 196 3(1) 8(3) 196 3(1) 6(2) 196 1(1) 4(2) 197 6(1) 8(2) 203 1(1) 6(2) 7 7(3) 2(1) 4 7(2) 2(1) 4 8(3) 3(1) 13 8(3) 2(1) 1 6(2) 1(1) 15 4(2) 2(1) 6 8(2) 3(1) 12 6(2) 3(1) 14 8(3) 3(1) 13 9(2) 2(1) 16 7(2) 3(1) 22 5(2) 3(1) 19 7(2) 1(1) 50 9(2) 1(1) 50 8(2) 5(1) 19 2(1) 4(2) 72 9(2) 1(1) 34 6(2) 3(1) 62 8(2) 4(1) 58 9(2) 4(1) 27 7(2) 5(1) 77 9(2) 1(1) 46 9(2) 1(1) 66 1(1) 7(2) 87 9(2) 2(1) 42 9(3) 2(1) 88 9(2) 1(1) 64 5(2) 2(1) 71 8(2) 4(1) 92 5(1) 8(2) 45 7(2) 5(1) 113 1(1) 9(2) 86 3(1) 5(2) 72 5(2) 3(1) 95 4(1) 8(2) 62 5(1) 7(2) 127 2(1) 9(2) 123 1(1) 9(2) 84 9(2) 4(1) 117 1(1) 6(2) 86 9(3) 1(1) 130 1(1) 9(2) 128 4(2) 2(1) 112 4(1) 8(2) 132 1(1) 6(2) 118 4(2) 1(1) 132 1(1) 9(2) 141 3(1) 6(2) 113 4(1) 7(2) 156 7(3) 1(1) 145 1(1) 9(3) 143 1(1) 9(2) 150 2(1) 9(3) 125 5(2) 2(1) 159 2(1) 7(2) 150 2(1) 4(2) 158 6(2) 1(1) 151 3(1) 4(2) 128 2(1) 5(2) 182 2(1) 7(2) 169 3(1) 7(2) 172 3(1) 5(2) 171 1(1) 9(2) 144 4(1) 9(2) 199 3(1) 9(3) 175 2(1) 8(3) 202 3(1) 6(2) 197 2(1) 6(2) 167 3(1) 7(3) 203 6(2) 3(1) 195 1(1) 8(3) 186 3(1) 5(2) 202 5(1) 8(2) 197 2(1) 9(3)
12 GPA GPA 4 0 GPA 1,2, : % 1 13% , ,
13 2.3: 1 GPA 2.4: 1 3 GPA 2.4 DA GPA 13
14 A, B GPA α, β, δ, γ GPA A 3.0 δ β α B 2.0 β γ α (A, α), (B, β) (A, β) A, B α, β (A, α), (B, β) (A, β), (B, α) = > 96.5 = A 3 α 2 β 1 2 GPA A 3.0 β( ) δ( ) α B 2.0 β γ α A (A, α), (B, β) (A, β), (B, α) = < = A 1,2,3 100,60,30 GPA A a G δ β α a 1 a 2 a 3 B b G β γ α b 1 b 2 b 3 14
15 A,B a 1,, a 3, b 1,, b 3 GPA a 1 > b 1 > a 2 > b 2 > a 3 > b 3 (A, α), (B, β) (A, β), (B, α) a 3 + b 1 > a 2 + b 3 a 1 = b 1 > a 2 = b 2 > a 3 = b 3 A GPA A a G β( ) δ( ) α ā 1 ā 2 a 3 B b G β γ α b 1 b 2 b 3 ā 1 > b 1 > ā 2 > b 2 > a 3 > b 3 ā 1 b 1 > a 3 b 3 A (A, α), (B, β) (A, β), (B, α) a 3 + b 1 < ā 1 + b 3 A 1 2 ā 1 = b 1 > ā 2 = b 2 > a 3 = b 3 DA yes DA DA
16 DA 4 GPA 2.7, : DA 1 d1 d2 d3 d4 d5 d6 d7 d8 d9 d : DA 2 25 d1 d2 d3 d4 d5 d6 d7 d8 d9 d d d3 d5 d
17 ,2,3 3 4,5,6 6 d d4,d DA [4] DA [9] A, B GPA α, β, δ, γ, ϵ GPA A 3.0 α β γ B 2.0 α δ ϵ α A,B 24 α 1 A,B 1 A 2 β A B 2 δ B 1 B δ (A, α), (B, ϵ) (A, β), (B, α) (A, α), (B, ϵ) (A, β), (B, α) = < =
18 2 1 2 A,B A 1 B DA [17, 16, 18] DA 3 3 GPA 18
19 [1] D. Gale and L.S. Shapley, College admissions and the stability of marriage, American Mathematical Monthly, Vol.69 (1962) [2] (1997). [3], :, 36 (1991) [4], :, TORSJ Vol.51 (2008) [5], :. OR 2005, (2005), [6] (2000). [7] :, 14.2 (2002) [8] :, 35 (2006) [9] :, 44 (2011) [10] :, (2013). [11] :, RAMP 25 (2013) [12] :, (2010) [13] NTT (2010). [14] :, RAMP 25 (2013) [15] :, RAMP 25 (2013) [16] Canadian Resident Matching Service, [17] National Resident Matching Program, [18], 19
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