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
20 The use of an optimization technique to solve student sectioning problems Keisuke Hotta Faculty of Business Administration, Bunkyo University Recieved: 30 April 2015 / Accepted: 25 May 2015 Abstract The focus of this study is to examine typical student sectioning problems. All students have their preferences to each section. The goal is to make an assignment decision to comply with their wishes as much as possible. To solve this problem, various approaches and algorithms have been proposed and discussed. In the field of mechanism design, both the deferred acceptance system and the top trading cycles system are considered to be a good method. It is studied well whether a solution provided by each method has desirable properties such as Pareto optimality, stability and strategy-proofness. In contrast, approaches using an optimization model are also studied. Studies of the model mainly focus on how to express student preferences in order to get a desirable solution. This study shows that the optimization approach is superior to the deferred acceptance to deal with some problems of student sectioning at a university. In addition, this research will clarify whether the solution by the optimization model has the desirable properties such as stability and strategy-proofness. Keyword: student sectioning problem, optimization, Boston public schools system, deferred acceptance system, top trading cycles system, Gale-Shapley algorithm, mechanism design, Pareto optimality, stability, strategy-proofness Faculty of Business Administration, Bunkyo University 1100 Namegaya, Chigasaki, Kanagawa , JAPAN Tel , Fax
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