A Reduced Semidefinite Programming Formulation for HA Assignment Problems in Sport Scheduling

LARA, H. J.; SIQUEIRA, A.S.; YUAN, J.

doi:10.5540/tema.2018.019.03.0471

H. J. LARA

Departamento de Engenharia, Universidade Federal de Santa Catarina-Blumenau, Rua João Pessoa, 2750, 89036-256, Blumenau, SC, Brazil. E-mail: hugol@ucla.edu.ve

A.S. SIQUEIRA

Departamento de Matemática, Universidade Federal do Parana, Curitiba, Brazil. E-mail: abel.s.siqueira@gmail.com; yuanjy@gmail.com

J. YUAN

Departamento de Matemática, Universidade Federal do Parana, Curitiba, Brazil. E-mail: abel.s.siqueira@gmail.com; yuanjy@gmail.com

*Corresponding author: Hugo José Lara - E-mail: hugol@ucla.edu.ve

n	Fixing FO			Fixing timetable
n	QLC	QQC	Deviation	QLC	QQC	Deviation
2	835.5	957.7	0.12760	840.2	886,4	0.05212
3	2148.5	2395	0.10292	1845.3	2231	0.17288
4	3244.9	3727.9	0.12956	3332.5	3925,8	0.15113
5	5479.9	6205.8	0.11697	5364.0	6110,8	0.12221
6	7334.4	8730.1	0.15987	7672.3	9263,6	0.17178
7	10380.1	12089.2	0.14137	10757.9	12433,7	0.13478
8	13779.5	15912.5	0.13405	13799.5	15893,6	0.13176
9	16792.9	19349.2	0.13211	17531.8	20705,9	0.15329
10	21204.6	24875.6	0.14757	21869.0	25935,2	0.15678

n	QLC fixing timetable
	full		reduced
	mean	SD	mean	SD
2	0.02768	0.01198	0.02247	0.01144
3	0.09818	0.01719	0.10523	0.03064
4	0.29231	0.07935	0.32632	0.12838
5	1.14851	0.29057	1.17092	0.30648
6	6.92692	0.74739	6.68803	1.08223
7	10.4368	0.99115	9.95315	0.89028
8	15.1239	1.65025	13.8831	1.45535
9	20.6103	1.86102	21.6154	7.18129
10	50.1043	21.6128	60.6731	22.7746

n	QLC fixing timetable
	full		reduced
	mean	SD	mean	SD
2	0.01881	0.00349	0.02049	0.00817
3	0.10556	0.03898	0.09833	0.02711
4	0.22068	0.06098	0.24719	0.07874
5	1.02054	0.30153	1.15223	0.54655
6	6.27446	1.73542	6.07395	1.71097
7	10.9502	1.57499	9.91357	0.59958
8	14.5375	1.38106	13.5211	1.30986
9	21.3364	1.86914	21.9286	6.80267
10	60.8347	19.6424	41.4042	22.7594

n	Fixing OF		Fixing Timetable
n	mean	sd	mean	sd
2	0.02107	0.00155	0.02079	0.00139
3	0.03974	0.00286	0.03858	0.00298
4	0.06538	0.00474	0.06054	0.00273
5	0.10066	0.00444	0.10122	0.00608
6	0.15408	0.00958	0.14684	0.00681
7	0.22715	0.00974	0.21647	0.00427
8	0.30999	0.00893	0.30643	0.00908
9	0.41958	0.01458	0.42272	0.01283
10	0.56785	0.00973	0.57336	0.00966

n	Linear		SDP MRC		Reduced SDP
n	$\bar{f}$	$\bar{t}$	${\bar{E}}_{f}$	$\bar{t}$	${\bar{E}}_{f}$	$\bar{t}$
2	7.3300e+02	0.02	-1.2258e-01	1.10	0.0000e+00	0.44
3	1.7891e+03	0.05	-2.8128e-01	3.95	0.0000e+00	0.27
4	3.2777e+03	0.15	-3.0212e-02	13.81	8.7450e-03	0.64
5	5.1840e+03	0.29	7.8695e-03	38.74	9.7976e-03	1.47
6	7.7355e+03	0.48	1.1984e-02	113.39	1.6149e-02	3.13
7	1.0212e+04	0.87	3.3375e-02	253.88	3.2369e-02	5.33
8	1.3432e+04	3.16	3.5014e-02	516.90	5.3592e-02	9.33
9	1.6894e+04	5.11	2.8511e-02	1070.48	4.4611e-02	16.47
10	2.0920e+04	7.46	8.2483e-02	2065.31	8.0239e-02	29.03
11	2.5079e+04	13.24	6.5871e-02	3600.69	6.0877e-02	49.06
12	2.8963e+04	17.38	8.8370e-02	5487.36	7.5394e-02	75.73
13	3.4313e+04	25.94	9.5084e-02	8925.42	9.8324e-02	114.65
14	4.0464e+04	91.94	1.3480e-01	13786.32	1.3321e-01	180.81
15	4.5725e+04	97.57	1.1949e-01	21204.91	1.1993e-01	229.93
16	5.3133e+04	172.20	1.3060e-01	34686.28	1.2608e-01	363.37
17	6.0957e+04	674.33	1.1399e-01	46958.26	1.1160e-01	731.70
18	6.6762e+04	1013.89	1.3102e-01	67643.59	1.2555e-01	819.57
19	7.5247e+04	1666.63	1.3078e-01	104376.27	1.3800e-01	1039.97
20	8.0043e+04	1104.60	1.4696e-01	123204.42	1.4079e-01	1459.20

\begin{array}{l} Figure 1: A timetable matrix for n = 2 \\ T = \begin{matrix} 2 & 3 & 2 & 4 & 3 & 4 \\ 1 & 4 & 1 & 3 & 4 & 3 \\ 4 & 1 & 4 & 2 & 1 & 2 \\ 3 & 2 & 3 & 1 & 2 & 1 \end{matrix} \end{array}

\begin{array}{l} Figure 2: An HA-assignment matrix consistent with 𝒯 \\ 𝒜= \begin{matrix} H & H & A & A & A & H \\ A & H & H & H & A & A \\ A & A & H & A & H & H \\ H & A & A & H & H & A \end{matrix} \end{array}

l (t, s) = {\begin{matrix} d (t, t) = 0 \begin{matrix}  \end{matrix} if (a_{t s}, a_{t s + 1}) = (H, H) \\ d (τ (t, s), τ (t, s + 1)) \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} if (a_{t s}, a_{t s + 1}) = (A, A) \\ d (t, τ (t, s + 1)) \begin{matrix}  \end{matrix} if (a_{t s}, a_{t s + 1}) = (H, A) \\ d (τ (t, s), t) \begin{matrix}  \end{matrix} if (a_{t s}, a_{t s + 1}) = (A, H) . \end{matrix}

\begin{array}{l} minimize l (x) = \frac{1}{4} \sum_{u, v \in V} w_{u v} (1 - x_{u} x_{v}) \\ subject to \begin{matrix}  \end{matrix} x_{v}^{2} = 1 \begin{matrix}  \end{matrix} \forall v \in V \\ \begin{matrix}  \end{matrix} x_{u} x_{v} = - 1 \begin{matrix}  \end{matrix} \forall {u, v} \in E_{c u t} \\ \begin{matrix}  \end{matrix} x_{r} = 1. \end{array}

\begin{array}{l} minimize \begin{matrix}  \end{matrix} l (X) = 〈 C, X 〉 \\ subject to \begin{matrix}  \end{matrix} X_{v v} = 1 \begin{matrix}  \end{matrix} \forall v \in V \\ \begin{matrix}  \end{matrix} X_{u v} = - 1 \begin{matrix}  \end{matrix} \forall {u, v} \in E_{c u t} \\ \begin{matrix}  \end{matrix} X \underline{≻} 0. \end{array}

\begin{matrix} l (t, s) = d (t', t ") | v_{t s} \cap V' | | v_{t s + 1} \cap V' | + d (t, t ") (1 - | v_{t s} \cap V' |) | v_{t s + 1} \cap V' | \\ + d (t', t) | v_{t s} \cap V' | (1 - | v_{t s + 1} \cap V' |) \end{matrix}

\begin{matrix} | v_{t s} \cap V' | | v_{t s + 1} \cap V' | = \frac{1}{2} (| {r, v_{t s}} \cap δ (V') | + | {r, v_{s + 1}} \cap δ (V') | \\ - | {v_{t s}, v_{s + 1}} \cap δ (V') |) . \end{matrix}

\begin{array}{l} Minimize \begin{matrix}  \end{matrix} l (X) = 〈 C, X 〉 \\ subject to \begin{matrix}  \end{matrix} X_{v v} = 1 \begin{matrix}  \end{matrix} v \in V 1 \\ \begin{matrix}  \end{matrix} X_{v w} = - 1 \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} {v, w} \in E 1_{c u t} \\ \begin{matrix}  \end{matrix} X \underline{≻} 0 \end{array}

\begin{array}{l} Figure 3: Labels assigned to each component on the timetable \\ γ = \begin{matrix} [1,1] & [2,1] & [1,2] & [3,1] & [2,2] & [3,2] \\ [1,3] & [4,1] & [1,4] & [5,1] & [4,2] & [5,2] \\ [6,1] & [2,3] & [6,2] & [5,3] & [2,4] & [5,4] \\ [6,3] & [4,3] & [6,4] & [3,3] & [4,4] & [3,4] \end{matrix} \end{array}

\begin{array}{l} X_{(t_{k} s_{k}) (t_{k} s_{k}^{'})} = - 1 \\ X_{(t_{k} s_{k}) (t_{k}^{'} s_{k})} = - 1 \\ X_{(t_{k} s_{k}^{'}) (t_{k}^{'} s_{k}^{'})} = - 1 \\ X_{(t_{k}^{'} s_{k}) (t_{k}^{'} s_{k}^{'})} = - 1, \end{array}

\begin{array}{l} l (t, s) = d (t', t ") y_{t s} y_{t s + 1} + d (t, t ") (1 - y_{t s}) y_{t s + 1} + d (t', t) y_{t s} (1 - y_{t s + 1}) \\ = d (t, t ") y_{t s + 1} + d (t', t) y_{t s} + (d (t', t ") - d (t, t ") - d (t', t)) y_{t s} y_{t s + 1} \end{array}

\begin{array}{l} (2 y_{t_{k} s_{k}} - 1) (2 y_{t_{k} s_{k}^{'}} - 1) = - 1 \\ (2 y_{t_{k} s_{k}} - 1) (2 y_{t_{k}^{'} s_{k}} - 1) = - 1 \\ (2 y_{t_{k} s_{k}^{'}} - 1) (2 y_{t_{k}^{'} s_{k}^{'}} - 1) = - 1 \\ (2 y_{t_{k}^{'} s_{k}} - 1) (2 y_{t_{k}^{'} s_{k}^{'}} - 1) = - 1, \end{array}

\begin{array}{l} y_{t_{k} s_{k}} + y_{t_{k} s_{k}^{'}} - 2 y_{t_{k} s_{k}} y_{t_{k} s_{k}^{'}} = 1 \\ y_{t_{k} s_{k}} + y_{t_{k}^{'} s_{k}} - 2 y_{t_{k} s_{k}} y_{t_{k}^{'} s_{k}} = 1 \\ y_{t_{k} s_{k}^{'}} + y_{t_{k}^{'} s_{k}^{'}} - 2 y_{t_{k} s_{k}^{'}} y_{t_{k}^{'} s_{k}^{'}} = 1 \\ y_{t_{k}^{'} s_{k}} + y_{t_{k}^{'} s_{k}^{'}} - 2 y_{t_{k}^{'} s_{k}} y_{t_{k}^{'} s_{k}^{'}} = 1 \end{array}

\begin{array}{l} minimize l (y) = 〈 (\begin{matrix} a & \frac{1}{2} c^{T} \\ \frac{1}{2} c & \frac{1}{2} Q \end{matrix}), (\begin{matrix} 1 & y^{T} \\ y & y y^{T} \end{matrix}) 〉 \\ subject to 〈 (\begin{matrix} 1 & \frac{1}{2} c_{k j}^{T} \\ \frac{1}{2} c_{k j} & \frac{1}{2} H_{k j} \end{matrix}), (\begin{matrix} 1 & y^{T} \\ y & y y^{T} \end{matrix}) 〉 = 0 \begin{matrix}  \end{matrix} \begin{matrix} k = 1,..., n (2 n - 1); \\ j = 1,2,3,4. \end{matrix} \\ \begin{matrix}  \end{matrix} y \in {0,1}^{2 n (4 n - 2)} . \end{array}

\begin{array}{l} minimize l (y) = 〈 (\begin{matrix} a & \frac{1}{2} c^{T} \\ \frac{1}{2} c & \frac{1}{2} Q \end{matrix}), Y 〉 \\ subject to 〈 (\begin{matrix} 1 & \frac{1}{2} c_{k j}^{T} \\ \frac{1}{2} c_{k j} & \frac{1}{2} H_{k j} \end{matrix}), Y 〉 = 0, \begin{matrix}  \end{matrix} \begin{matrix} k = 1,..., n (2 n - 1); \\ j = 1,2,3,4. \end{matrix} \\ \begin{matrix}  \end{matrix} Y_{(t s) (t s)} - Y_{(11) (t s)} = 0, \begin{matrix} \forall (t, s) \in T \times S \end{matrix} \\ \begin{matrix}  \end{matrix} Y \underline{≻} 0. \end{array}

\begin{array}{l} y_{t_{k} s_{k}} + y_{t_{k}^{'} s_{k}} = 1 \\ y_{t_{k} s_{k}} + y_{t_{k} s_{k}^{'}} = 1 \\ - y_{t_{k} s_{k}} + y_{t_{k}^{'} s_{k}} = 0 \end{array}

[\begin{matrix} y_{t_{k}^{'} s_{k}} \\ y_{t_{k} s_{k}^{'}} \\ y_{t_{k}^{'} s_{k}^{'}} \end{matrix}] = [\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}] - [\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}] y_{t_{k} s_{k}} = b_{k} - A_{t_{k} s_{k}} y_{t_{k} s_{k}} .

[1] *Corresponding author: Hugo José Lara - E-mail: hugol@ucla.edu.ve

Brasil

Brasil

A Reduced Semidefinite Programming Formulation for HA Assignment Problems in Sport Scheduling

ABSTRACT