estadística
ciencia de datos
2025
Author

José Luis Cañadas Reche

Published

July 20, 2025

Listening

Este post viene a ampliar el de la IO al servicio del mal. y está en Work in Progress.

Show the code 
using Pkg
Pkg.activate(@__DIR__)
Pkg.instantiate()
using JuMP, DataFrames, CSV, Plots, HiGHS
  Activating project at `~/proyectos_personales/blog_quarto/2025/07`
Show the code 


ANDALUCIA = ["ALMERIA", "CADIZ",
 "CORDOBA","GRANADA",
"HUELVA", "JAEN",
"MALAGA", 
"SEVILLA"]

cod_postales_raw = DataFrame(CSV.File("cod_postales.csv"))
sedes_raw = DataFrame(CSV.File("sedes.csv"))

cod_postales_raw
10808×6 DataFrame
10783 rows omitted
Row cod_postal cod_prov area_m2 centroide_longitud centroide_latitud provincia
Int64 Int64 Float64 Float64 Float64 String31
1 35560 35 1.87875e8 -13.703 29.0501 PALMAS, LAS
2 27330 27 6.65941e6 -7.3956 42.5614 LUGO
3 46680 46 6.91908e7 -0.429241 39.2137 VALENCIA/VALENCIA
4 49706 49 9.02291e7 -5.726 41.3027 ZAMORA
5 21120 21 2.00686e7 -6.99673 37.2879 HUELVA
6 16623 16 1.3286e8 -2.35389 39.6706 CUENCA
7 10650 10 8.92692e7 -6.20114 40.1693 CACERES
8 37111 37 1.01925e8 -5.89406 41.1513 SALAMANCA
9 44124 44 4.63729e7 -1.55171 40.3196 TERUEL
10 33150 33 1.57172e6 -6.14814 43.5619 ASTURIAS
11 41910 41 2.33707e6 -6.0401 37.3883 SEVILLA
12 28597 28 1.03083e8 -3.15911 40.111 MADRID
13 3838 3 3.26148e7 -0.563752 38.7809 ALICANTE/ALACANT
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
10797 45635 45 3.22399e7 -4.84787 40.0932 TOLEDO
10798 36859 36 8.31546e6 -8.54405 42.4389 PONTEVEDRA
10799 37554 37 3.34345e7 -6.81653 40.4588 SALAMANCA
10800 7190 7 7.04852e7 2.58113 39.6665 BALEARS, ILLES
10801 8280 8 1.65922e7 1.50624 41.7216 BARCELONA
10802 27146 27 4.13968e7 -7.43692 43.0169 LUGO
10803 45006 45 3.70004e7 -3.99207 39.8375 TOLEDO
10804 47493 47 1.07702e8 -4.86432 41.2334 VALLADOLID
10805 50547 50 1.75265e8 -1.51964 41.697 ZARAGOZA
10806 27460 27 1.43119e7 -7.58225 42.4526 LUGO
10807 28921 28 9.61587e5 -3.8276 40.3477 MADRID
10808 14140 14 3.0837e7 -4.8546 37.6964 CORDOBA

Seleccionamos solo Andalucía. Tanto para las sedes como para las códigos postales

Show the code 

cod_postales = cod_postales_raw[[i ∈ ANDALUCIA for i in cod_postales_raw.provincia ], :]

sedes = subset(sedes_raw, :provincia => ByRow(x -> x ∈ ANDALUCIA))
76×7 DataFrame
51 rows omitted
Row cod_postal tipo cod_prov area_m2 centroide_longitud centroide_latitud provincia
Int64 String7 Int64 Float64 Float64 Float64 String31
1 4727 jefe 4 9.58884e6 -2.62205 36.8347 ALMERIA
2 11600 jefe 11 1.08706e8 -5.46371 36.6586 CADIZ
3 14659 jefe 14 2.76092e7 -4.42429 37.9386 CORDOBA
4 18328 jefe 18 5.85745e7 -3.87328 37.1906 GRANADA
5 21387 jefe 21 2.19028e7 -6.69674 38.0752 HUELVA
6 23369 jefe 23 1.2085e8 -2.80429 38.3832 JAEN
7 29531 jefe 29 1.24176e8 -4.74224 37.1056 MALAGA
8 41760 jefe 41 1.44046e8 -5.61951 37.0264 SEVILLA
9 41599 jefe 41 8.47675e7 -4.72387 37.2476 SEVILLA
10 14729 jefe 14 1.71617e8 -4.99471 37.855 CORDOBA
11 18006 jefe 18 3.24591e6 -3.60903 37.1606 GRANADA
12 14547 jefe 14 1.88571e8 -4.83313 37.5774 CORDOBA
13 14430 jefe 14 5.04828e8 -4.57405 38.0962 CORDOBA
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
65 11611 indio 11 9.26568e7 -5.36712 36.6863 CADIZ
66 14913 indio 14 4.36996e7 -4.48437 37.2767 CORDOBA
67 21387 indio 21 2.19028e7 -6.69674 38.0752 HUELVA
68 14913 indio 14 4.36996e7 -4.48437 37.2767 CORDOBA
69 4727 indio 4 9.58884e6 -2.62205 36.8347 ALMERIA
70 11406 indio 11 2.57144e7 -6.10322 36.6714 CADIZ
71 14659 indio 14 2.76092e7 -4.42429 37.9386 CORDOBA
72 14191 indio 14 2.36389e7 -4.89704 37.7 CORDOBA
73 4727 indio 4 9.58884e6 -2.62205 36.8347 ALMERIA
74 29580 indio 29 8.54712e7 -4.62152 36.7579 MALAGA
75 21594 indio 21 1.55329e8 -7.42947 37.5182 HUELVA
76 29716 indio 29 4.602e7 -4.08153 36.8767 MALAGA

Ordenamos el dataset de sedes por si son jefes o indios

Show the code 
sort!(sedes, :tipo, rev = true)
76×7 DataFrame
51 rows omitted
Row cod_postal tipo cod_prov area_m2 centroide_longitud centroide_latitud provincia
Int64 String7 Int64 Float64 Float64 Float64 String31
1 4727 jefe 4 9.58884e6 -2.62205 36.8347 ALMERIA
2 11600 jefe 11 1.08706e8 -5.46371 36.6586 CADIZ
3 14659 jefe 14 2.76092e7 -4.42429 37.9386 CORDOBA
4 18328 jefe 18 5.85745e7 -3.87328 37.1906 GRANADA
5 21387 jefe 21 2.19028e7 -6.69674 38.0752 HUELVA
6 23369 jefe 23 1.2085e8 -2.80429 38.3832 JAEN
7 29531 jefe 29 1.24176e8 -4.74224 37.1056 MALAGA
8 41760 jefe 41 1.44046e8 -5.61951 37.0264 SEVILLA
9 41599 jefe 41 8.47675e7 -4.72387 37.2476 SEVILLA
10 14729 jefe 14 1.71617e8 -4.99471 37.855 CORDOBA
11 18006 jefe 18 3.24591e6 -3.60903 37.1606 GRANADA
12 14547 jefe 14 1.88571e8 -4.83313 37.5774 CORDOBA
13 14430 jefe 14 5.04828e8 -4.57405 38.0962 CORDOBA
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
65 11611 indio 11 9.26568e7 -5.36712 36.6863 CADIZ
66 14913 indio 14 4.36996e7 -4.48437 37.2767 CORDOBA
67 21387 indio 21 2.19028e7 -6.69674 38.0752 HUELVA
68 14913 indio 14 4.36996e7 -4.48437 37.2767 CORDOBA
69 4727 indio 4 9.58884e6 -2.62205 36.8347 ALMERIA
70 11406 indio 11 2.57144e7 -6.10322 36.6714 CADIZ
71 14659 indio 14 2.76092e7 -4.42429 37.9386 CORDOBA
72 14191 indio 14 2.36389e7 -4.89704 37.7 CORDOBA
73 4727 indio 4 9.58884e6 -2.62205 36.8347 ALMERIA
74 29580 indio 29 8.54712e7 -4.62152 36.7579 MALAGA
75 21594 indio 21 1.55329e8 -7.42947 37.5182 HUELVA
76 29716 indio 29 4.602e7 -4.08153 36.8767 MALAGA

Creamos funcion haversine que nos va a dar la distancia entre los puntos dada longitud y latitud. Hay que pasarle las longitudes y latitude y el radio estimado de la tierra.

Show the code 

"""
    haversine(lat1, long1, lat2, long2, r = 6372.8)

Compute the haversine distance between two points on a sphere of radius `r`,
where the points are given by the latitude/longitude pairs `lat1/long1` and
`lat2/long2` (in degrees).
"""
function haversine(lat1, long1, lat2, long2, r = 6372.8)
    lat1, long1 = deg2rad(lat1), deg2rad(long1)
    lat2, long2 = deg2rad(lat2), deg2rad(long2)
    hav(a, b) = sin((b - a) / 2)^2
    inner_term = hav(lat1, lat2) + cos(lat1) * cos(lat2) * hav(long1, long2)
    d = 2 * r * asin(sqrt(inner_term))
    # Round distance to nearest kilometer.
    return round( d, digits = 2)
end
Main.Notebook.haversine

y ahora calculamos las distancias entre todos los códigos postales y todas las sedes de los trabajadores. En Julia esto es muy rápido

Show the code 
# number of clients
n = size(cod_postales,1)
# number of  comerciales (facilities)
m = size(sedes, 1)

# esto es como en python, una list comprehension

c  = [haversine( 
           cod_postales.centroide_latitud[i], cod_postales.centroide_longitud[i],
           sedes.centroide_latitud[j], sedes.centroide_longitud[j]) 
           for i in 1:n, j in 1:m]



1199×76 Matrix{Float64}:
 391.51  153.15  237.92  276.78   91.46  …  391.51  219.0    46.02  262.68
 309.35   95.96  154.88  192.99   95.77     309.35  144.1   123.52  182.75
 237.91   17.41  165.47  140.3   205.33     237.91   60.56  215.89  110.43
 160.03   93.98  134.67   70.39  251.29     160.03   18.83  281.53   33.89
 133.54  260.16  121.23  114.95  320.21     133.54  195.38  389.21  153.88
 283.0    38.8   169.44  174.7   154.78  …  283.0   105.93  160.81  153.04
 109.45  162.65  100.43    9.65  275.83     109.45   90.71  325.81   45.8
 188.54   89.03   88.14   73.36  202.19     188.54   47.17  244.21   64.36
  64.33  307.66  222.48  159.52  418.35      64.33  231.99  474.69  182.17
 150.59  295.03  150.67  149.44  347.02     150.59  230.27  419.08  188.03
   ⋮                                     ⋱                            ⋮
 201.53  102.74   67.24   83.61  184.43  …  201.53   73.17  234.3    85.23
   9.86  263.86  208.24  126.87  393.96       9.86  188.12  442.11  139.72
 125.69  194.52   77.57   52.75  273.43     125.69  130.18  334.64   92.18
 311.14   57.82  203.18  206.25  161.74     311.14  132.87  148.4   181.79
 165.85  300.7   150.32  158.46  343.75     165.85  238.11  417.88  197.55
 367.22  164.67  189.14  249.28   27.68  …  367.22  211.0    83.48  245.41
   0.0   254.01  201.1   117.96  385.28       0.0   178.29  432.72  129.97
 192.17   92.83   81.27   75.61  196.58     192.17   55.23  241.0    70.37
 219.63  127.43   46.43  103.32  167.11     219.63  106.41  227.74  113.98

Ahora para hacer truco de que a los jefes les toque códigos postales más cercanos a su sede lo que hacemos es aumentar la distancia artificialmente.

Show the code 
# utilizamos operador ternario 
# Si eres jefe te pongo distancia 1.1, si no 0.9
for j in 1:m
   c[:,j] = sedes.tipo[j] == "jefe" ? 1.1 * c[:, j] : 0.9 * c[:,j]
end

Contamos cuantos jefes y cuantos indios hay.

Show the code 
n_jefes = sum(sedes.tipo .=="jefe")
print(n_jefes)
27
Show the code 
n_indios = m - n_jefes
print(n_indios)
49

Y ahora definimos el problema de optimización y sus restricciones

Show the code 
# Fixed costs
f = ones(m);

Definimos el modelo dónde x va a ser una matriz de variables binarias qeu va a indicar si un código postal (de 1 a n) se asigna a un comercial (jefe o indio) va de 1 a m. Y la variable y va a indicar si ese código postal (o tienda) está asignado o no. Si no está asignado entonces se podrá asignar a un comercial.

Show the code 
ufl = Model(HiGHS.Optimizer)

@variable(ufl, x[1:n, 1:m], Bin);
@variable(ufl, y[1:m], Bin);

Ahora vamos a poner restricione está asignado o no. Si no está asignado entonces se podrá asignar a un comercial.

Show the code 
ufl = Model(HiGHS.Optimizer)

@variable(ufl, x[1:n, 1:m], Bin);
@variable(ufl, y[1:m], Bin);

Ahora vamos a poner restriciones

Lo primero es definir una serie de restricciones de forma que a cada código postal sólo se le puede asignar un comercial. Eso se define haciendo que en la matriz X , que tiene n filas (sedes) y j columnas (comercials) , al sumar por fila el valor sea igual a 1. Es decir, a cada código postal sólo puede ir un comercial. (Se podría cambiar a que fueran 2 por ejemplo)

Show the code 
# esta restriccion es poner que  a  cada  cod postal solo puede ir un comecial, (sea indio o jefe)

@constraint(ufl, client_service[i in 1:n], sum(x[i, j] for j in 1:m) == 1);

La siguiente restricción es poner que un comercial sólo puede ser asignado a un código postal (sede, tienda) que esté abierta.

Esto se hace haciendo que la matriz x[i,j] sea menor o igual que y[j]. puesto que y[j] vale 1 si la “tienda” está abierta, y por tanto si está cerrada vale 0 y x[i,j] por narices tiene que ser 0 , es decir, si la “tienda” ha sido asignada ya no se puede

Show the code 
# siguiente restricción es poner que un cod postales solo puede ir a una que esté abierta
# esto es, si y[j] = 0 significa sede cerrada y no se puede asignar nadie ahí

@constraint(ufl, open_facility[i in 1:n, j in 1:m], x[i, j] <= y[j]);

Estas dos restricciones implican. - Cada código postal sólo puede ser visitado por un único comercial - Un comercial sólo puede ser asignado a una sede si esta está abierta

Si tengo 3 cod postales y 2 comerciales

sum(x[1,1], x[1,2]) == 1  # cod postal 1 → un solo comercial
sum(x[2,1], x[2,2]) == 1  # cod postal 2 → un solo comercial
sum(x[3,1], x[3,2]) == 1  # cod postal 3 → un solo comercial

x[i,j] <= y[j]            # Solo puedes usar comercial j si está abierto

De esta forma, un comercial puede ir a varios códigos postales pero un código postal solo puede ser visitado por un comercial.

Ahora vamos con las restriciones diferenciadas según el tipo de comercial.

La restricción para los indios es que al menos tienen que visitar 3 códigos postales cada uno y como máximo 100. Mientras que para los jefes se impone que al menos tienen que visitar un código postal (si no, sería muy cantoso) y cómo máximo 10.

Show the code 

# restricciones para los indios, jefes  
@constraint(ufl, tot_por_indio[j in (n_jefes+1):m], 3 <= sum(x[i, j] for i in 1:n) <= 100);
@constraint(ufl, tot_por_jefe[j ∈ 1:n_jefes], 1<=sum(x[i, j] for i in 1:n) <= 10);

Si hacemos print(ufl) veremos las restricciones planteadas, pero son muchas.

Y con esto ya podemos plantear el objetivo. que va a ser

Show the code 
@objective(ufl, Min, f'y + sum(c .* x));

En Jump también podemos escribir la formulación en LaTex

Show the code 
latex_formulation(ufl)
$$ \begin{aligned}
\min\quad & y_{1} + y_{2} + y_{3} + y_{4} + y_{5} + y_{6} + y_{7} + y_{8} + y_{9} + y_{10} + y_{11} + y_{12} + y_{13} + y_{14} + y_{15} + y_{16} + y_{17} + y_{18} + y_{19} + y_{20} + y_{21} + y_{22} + y_{23} + y_{24} + y_{25} + y_{26} + y_{27} + y_{28} + y_{29} + y_{30} + [[\ldots\text{91064 terms omitted}\ldots]] + 186.957 x_{1170,76} + 253.81799999999998 x_{1171,76} + 150.07500000000002 x_{1172,76} + 53.352000000000004 x_{1173,76} + 96.066 x_{1174,76} + 162.18 x_{1175,76} + 161.39700000000002 x_{1176,76} + 80.82 x_{1177,76} + 87.507 x_{1178,76} + 78.68700000000001 x_{1179,76} + 209.214 x_{1180,76} + 146.025 x_{1181,76} + 165.366 x_{1182,76} + 156.89700000000002 x_{1183,76} + 86.36399999999999 x_{1184,76} + 96.696 x_{1185,76} + 34.74 x_{1186,76} + 122.463 x_{1187,76} + 93.222 x_{1188,76} + 154.746 x_{1189,76} + 165.123 x_{1190,76} + 76.70700000000001 x_{1191,76} + 125.748 x_{1192,76} + 82.962 x_{1193,76} + 163.611 x_{1194,76} + 177.79500000000002 x_{1195,76} + 220.869 x_{1196,76} + 116.973 x_{1197,76} + 63.333000000000006 x_{1198,76} + 102.58200000000001 x_{1199,76}\\
\text{Subject to} \quad & x_{1,1} + x_{1,2} + x_{1,3} + x_{1,4} + x_{1,5} + x_{1,6} + x_{1,7} + x_{1,8} + x_{1,9} + x_{1,10} + x_{1,11} + x_{1,12} + x_{1,13} + x_{1,14} + x_{1,15} + x_{1,16} + x_{1,17} + x_{1,18} + x_{1,19} + x_{1,20} + x_{1,21} + x_{1,22} + x_{1,23} + x_{1,24} + x_{1,25} + x_{1,26} + x_{1,27} + x_{1,28} + x_{1,29} + x_{1,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{1,47} + x_{1,48} + x_{1,49} + x_{1,50} + x_{1,51} + x_{1,52} + x_{1,53} + x_{1,54} + x_{1,55} + x_{1,56} + x_{1,57} + x_{1,58} + x_{1,59} + x_{1,60} + x_{1,61} + x_{1,62} + x_{1,63} + x_{1,64} + x_{1,65} + x_{1,66} + x_{1,67} + x_{1,68} + x_{1,69} + x_{1,70} + x_{1,71} + x_{1,72} + x_{1,73} + x_{1,74} + x_{1,75} + x_{1,76} = 1\\
 & x_{2,1} + x_{2,2} + x_{2,3} + x_{2,4} + x_{2,5} + x_{2,6} + x_{2,7} + x_{2,8} + x_{2,9} + x_{2,10} + x_{2,11} + x_{2,12} + x_{2,13} + x_{2,14} + x_{2,15} + x_{2,16} + x_{2,17} + x_{2,18} + x_{2,19} + x_{2,20} + x_{2,21} + x_{2,22} + x_{2,23} + x_{2,24} + x_{2,25} + x_{2,26} + x_{2,27} + x_{2,28} + x_{2,29} + x_{2,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{2,47} + x_{2,48} + x_{2,49} + x_{2,50} + x_{2,51} + x_{2,52} + x_{2,53} + x_{2,54} + x_{2,55} + x_{2,56} + x_{2,57} + x_{2,58} + x_{2,59} + x_{2,60} + x_{2,61} + x_{2,62} + x_{2,63} + x_{2,64} + x_{2,65} + x_{2,66} + x_{2,67} + x_{2,68} + x_{2,69} + x_{2,70} + x_{2,71} + x_{2,72} + x_{2,73} + x_{2,74} + x_{2,75} + x_{2,76} = 1\\
 & x_{3,1} + x_{3,2} + x_{3,3} + x_{3,4} + x_{3,5} + x_{3,6} + x_{3,7} + x_{3,8} + x_{3,9} + x_{3,10} + x_{3,11} + x_{3,12} + x_{3,13} + x_{3,14} + x_{3,15} + x_{3,16} + x_{3,17} + x_{3,18} + x_{3,19} + x_{3,20} + x_{3,21} + x_{3,22} + x_{3,23} + x_{3,24} + x_{3,25} + x_{3,26} + x_{3,27} + x_{3,28} + x_{3,29} + x_{3,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{3,47} + x_{3,48} + x_{3,49} + x_{3,50} + x_{3,51} + x_{3,52} + x_{3,53} + x_{3,54} + x_{3,55} + x_{3,56} + x_{3,57} + x_{3,58} + x_{3,59} + x_{3,60} + x_{3,61} + x_{3,62} + x_{3,63} + x_{3,64} + x_{3,65} + x_{3,66} + x_{3,67} + x_{3,68} + x_{3,69} + x_{3,70} + x_{3,71} + x_{3,72} + x_{3,73} + x_{3,74} + x_{3,75} + x_{3,76} = 1\\
 & x_{4,1} + x_{4,2} + x_{4,3} + x_{4,4} + x_{4,5} + x_{4,6} + x_{4,7} + x_{4,8} + x_{4,9} + x_{4,10} + x_{4,11} + x_{4,12} + x_{4,13} + x_{4,14} + x_{4,15} + x_{4,16} + x_{4,17} + x_{4,18} + x_{4,19} + x_{4,20} + x_{4,21} + x_{4,22} + x_{4,23} + x_{4,24} + x_{4,25} + x_{4,26} + x_{4,27} + x_{4,28} + x_{4,29} + x_{4,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{4,47} + x_{4,48} + x_{4,49} + x_{4,50} + x_{4,51} + x_{4,52} + x_{4,53} + x_{4,54} + x_{4,55} + x_{4,56} + x_{4,57} + x_{4,58} + x_{4,59} + x_{4,60} + x_{4,61} + x_{4,62} + x_{4,63} + x_{4,64} + x_{4,65} + x_{4,66} + x_{4,67} + x_{4,68} + x_{4,69} + x_{4,70} + x_{4,71} + x_{4,72} + x_{4,73} + x_{4,74} + x_{4,75} + x_{4,76} = 1\\
 & x_{5,1} + x_{5,2} + x_{5,3} + x_{5,4} + x_{5,5} + x_{5,6} + x_{5,7} + x_{5,8} + x_{5,9} + x_{5,10} + x_{5,11} + x_{5,12} + x_{5,13} + x_{5,14} + x_{5,15} + x_{5,16} + x_{5,17} + x_{5,18} + x_{5,19} + x_{5,20} + x_{5,21} + x_{5,22} + x_{5,23} + x_{5,24} + x_{5,25} + x_{5,26} + x_{5,27} + x_{5,28} + x_{5,29} + x_{5,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{5,47} + x_{5,48} + x_{5,49} + x_{5,50} + x_{5,51} + x_{5,52} + x_{5,53} + x_{5,54} + x_{5,55} + x_{5,56} + x_{5,57} + x_{5,58} + x_{5,59} + x_{5,60} + x_{5,61} + x_{5,62} + x_{5,63} + x_{5,64} + x_{5,65} + x_{5,66} + x_{5,67} + x_{5,68} + x_{5,69} + x_{5,70} + x_{5,71} + x_{5,72} + x_{5,73} + x_{5,74} + x_{5,75} + x_{5,76} = 1\\
 & x_{6,1} + x_{6,2} + x_{6,3} + x_{6,4} + x_{6,5} + x_{6,6} + x_{6,7} + x_{6,8} + x_{6,9} + x_{6,10} + x_{6,11} + x_{6,12} + x_{6,13} + x_{6,14} + x_{6,15} + x_{6,16} + x_{6,17} + x_{6,18} + x_{6,19} + x_{6,20} + x_{6,21} + x_{6,22} + x_{6,23} + x_{6,24} + x_{6,25} + x_{6,26} + x_{6,27} + x_{6,28} + x_{6,29} + x_{6,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{6,47} + x_{6,48} + x_{6,49} + x_{6,50} + x_{6,51} + x_{6,52} + x_{6,53} + x_{6,54} + x_{6,55} + x_{6,56} + x_{6,57} + x_{6,58} + x_{6,59} + x_{6,60} + x_{6,61} + x_{6,62} + x_{6,63} + x_{6,64} + x_{6,65} + x_{6,66} + x_{6,67} + x_{6,68} + x_{6,69} + x_{6,70} + x_{6,71} + x_{6,72} + x_{6,73} + x_{6,74} + x_{6,75} + x_{6,76} = 1\\
 & x_{7,1} + x_{7,2} + x_{7,3} + x_{7,4} + x_{7,5} + x_{7,6} + x_{7,7} + x_{7,8} + x_{7,9} + x_{7,10} + x_{7,11} + x_{7,12} + x_{7,13} + x_{7,14} + x_{7,15} + x_{7,16} + x_{7,17} + x_{7,18} + x_{7,19} + x_{7,20} + x_{7,21} + x_{7,22} + x_{7,23} + x_{7,24} + x_{7,25} + x_{7,26} + x_{7,27} + x_{7,28} + x_{7,29} + x_{7,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{7,47} + x_{7,48} + x_{7,49} + x_{7,50} + x_{7,51} + x_{7,52} + x_{7,53} + x_{7,54} + x_{7,55} + x_{7,56} + x_{7,57} + x_{7,58} + x_{7,59} + x_{7,60} + x_{7,61} + x_{7,62} + x_{7,63} + x_{7,64} + x_{7,65} + x_{7,66} + x_{7,67} + x_{7,68} + x_{7,69} + x_{7,70} + x_{7,71} + x_{7,72} + x_{7,73} + x_{7,74} + x_{7,75} + x_{7,76} = 1\\
 & x_{8,1} + x_{8,2} + x_{8,3} + x_{8,4} + x_{8,5} + x_{8,6} + x_{8,7} + x_{8,8} + x_{8,9} + x_{8,10} + x_{8,11} + x_{8,12} + x_{8,13} + x_{8,14} + x_{8,15} + x_{8,16} + x_{8,17} + x_{8,18} + x_{8,19} + x_{8,20} + x_{8,21} + x_{8,22} + x_{8,23} + x_{8,24} + x_{8,25} + x_{8,26} + x_{8,27} + x_{8,28} + x_{8,29} + x_{8,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{8,47} + x_{8,48} + x_{8,49} + x_{8,50} + x_{8,51} + x_{8,52} + x_{8,53} + x_{8,54} + x_{8,55} + x_{8,56} + x_{8,57} + x_{8,58} + x_{8,59} + x_{8,60} + x_{8,61} + x_{8,62} + x_{8,63} + x_{8,64} + x_{8,65} + x_{8,66} + x_{8,67} + x_{8,68} + x_{8,69} + x_{8,70} + x_{8,71} + x_{8,72} + x_{8,73} + x_{8,74} + x_{8,75} + x_{8,76} = 1\\
 & x_{9,1} + x_{9,2} + x_{9,3} + x_{9,4} + x_{9,5} + x_{9,6} + x_{9,7} + x_{9,8} + x_{9,9} + x_{9,10} + x_{9,11} + x_{9,12} + x_{9,13} + x_{9,14} + x_{9,15} + x_{9,16} + x_{9,17} + x_{9,18} + x_{9,19} + x_{9,20} + x_{9,21} + x_{9,22} + x_{9,23} + x_{9,24} + x_{9,25} + x_{9,26} + x_{9,27} + x_{9,28} + x_{9,29} + x_{9,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{9,47} + x_{9,48} + x_{9,49} + x_{9,50} + x_{9,51} + x_{9,52} + x_{9,53} + x_{9,54} + x_{9,55} + x_{9,56} + x_{9,57} + x_{9,58} + x_{9,59} + x_{9,60} + x_{9,61} + x_{9,62} + x_{9,63} + x_{9,64} + x_{9,65} + x_{9,66} + x_{9,67} + x_{9,68} + x_{9,69} + x_{9,70} + x_{9,71} + x_{9,72} + x_{9,73} + x_{9,74} + x_{9,75} + x_{9,76} = 1\\
 & x_{10,1} + x_{10,2} + x_{10,3} + x_{10,4} + x_{10,5} + x_{10,6} + x_{10,7} + x_{10,8} + x_{10,9} + x_{10,10} + x_{10,11} + x_{10,12} + x_{10,13} + x_{10,14} + x_{10,15} + x_{10,16} + x_{10,17} + x_{10,18} + x_{10,19} + x_{10,20} + x_{10,21} + x_{10,22} + x_{10,23} + x_{10,24} + x_{10,25} + x_{10,26} + x_{10,27} + x_{10,28} + x_{10,29} + x_{10,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{10,47} + x_{10,48} + x_{10,49} + x_{10,50} + x_{10,51} + x_{10,52} + x_{10,53} + x_{10,54} + x_{10,55} + x_{10,56} + x_{10,57} + x_{10,58} + x_{10,59} + x_{10,60} + x_{10,61} + x_{10,62} + x_{10,63} + x_{10,64} + x_{10,65} + x_{10,66} + x_{10,67} + x_{10,68} + x_{10,69} + x_{10,70} + x_{10,71} + x_{10,72} + x_{10,73} + x_{10,74} + x_{10,75} + x_{10,76} = 1\\
 & x_{11,1} + x_{11,2} + x_{11,3} + x_{11,4} + x_{11,5} + x_{11,6} + x_{11,7} + x_{11,8} + x_{11,9} + x_{11,10} + x_{11,11} + x_{11,12} + x_{11,13} + x_{11,14} + x_{11,15} + x_{11,16} + x_{11,17} + x_{11,18} + x_{11,19} + x_{11,20} + x_{11,21} + x_{11,22} + x_{11,23} + x_{11,24} + x_{11,25} + x_{11,26} + x_{11,27} + x_{11,28} + x_{11,29} + x_{11,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{11,47} + x_{11,48} + x_{11,49} + x_{11,50} + x_{11,51} + x_{11,52} + x_{11,53} + x_{11,54} + x_{11,55} + x_{11,56} + x_{11,57} + x_{11,58} + x_{11,59} + x_{11,60} + x_{11,61} + x_{11,62} + x_{11,63} + x_{11,64} + x_{11,65} + x_{11,66} + x_{11,67} + x_{11,68} + x_{11,69} + x_{11,70} + x_{11,71} + x_{11,72} + x_{11,73} + x_{11,74} + x_{11,75} + x_{11,76} = 1\\
 & x_{12,1} + x_{12,2} + x_{12,3} + x_{12,4} + x_{12,5} + x_{12,6} + x_{12,7} + x_{12,8} + x_{12,9} + x_{12,10} + x_{12,11} + x_{12,12} + x_{12,13} + x_{12,14} + x_{12,15} + x_{12,16} + x_{12,17} + x_{12,18} + x_{12,19} + x_{12,20} + x_{12,21} + x_{12,22} + x_{12,23} + x_{12,24} + x_{12,25} + x_{12,26} + x_{12,27} + x_{12,28} + x_{12,29} + x_{12,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{12,47} + x_{12,48} + x_{12,49} + x_{12,50} + x_{12,51} + x_{12,52} + x_{12,53} + x_{12,54} + x_{12,55} + x_{12,56} + x_{12,57} + x_{12,58} + x_{12,59} + x_{12,60} + x_{12,61} + x_{12,62} + x_{12,63} + x_{12,64} + x_{12,65} + x_{12,66} + x_{12,67} + x_{12,68} + x_{12,69} + x_{12,70} + x_{12,71} + x_{12,72} + x_{12,73} + x_{12,74} + x_{12,75} + x_{12,76} = 1\\
 & x_{13,1} + x_{13,2} + x_{13,3} + x_{13,4} + x_{13,5} + x_{13,6} + x_{13,7} + x_{13,8} + x_{13,9} + x_{13,10} + x_{13,11} + x_{13,12} + x_{13,13} + x_{13,14} + x_{13,15} + x_{13,16} + x_{13,17} + x_{13,18} + x_{13,19} + x_{13,20} + x_{13,21} + x_{13,22} + x_{13,23} + x_{13,24} + x_{13,25} + x_{13,26} + x_{13,27} + x_{13,28} + x_{13,29} + x_{13,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{13,47} + x_{13,48} + x_{13,49} + x_{13,50} + x_{13,51} + x_{13,52} + x_{13,53} + x_{13,54} + x_{13,55} + x_{13,56} + x_{13,57} + x_{13,58} + x_{13,59} + x_{13,60} + x_{13,61} + x_{13,62} + x_{13,63} + x_{13,64} + x_{13,65} + x_{13,66} + x_{13,67} + x_{13,68} + x_{13,69} + x_{13,70} + x_{13,71} + x_{13,72} + x_{13,73} + x_{13,74} + x_{13,75} + x_{13,76} = 1\\
 & x_{14,1} + x_{14,2} + x_{14,3} + x_{14,4} + x_{14,5} + x_{14,6} + x_{14,7} + x_{14,8} + x_{14,9} + x_{14,10} + x_{14,11} + x_{14,12} + x_{14,13} + x_{14,14} + x_{14,15} + x_{14,16} + x_{14,17} + x_{14,18} + x_{14,19} + x_{14,20} + x_{14,21} + x_{14,22} + x_{14,23} + x_{14,24} + x_{14,25} + x_{14,26} + x_{14,27} + x_{14,28} + x_{14,29} + x_{14,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{14,47} + x_{14,48} + x_{14,49} + x_{14,50} + x_{14,51} + x_{14,52} + x_{14,53} + x_{14,54} + x_{14,55} + x_{14,56} + x_{14,57} + x_{14,58} + x_{14,59} + x_{14,60} + x_{14,61} + x_{14,62} + x_{14,63} + x_{14,64} + x_{14,65} + x_{14,66} + x_{14,67} + x_{14,68} + x_{14,69} + x_{14,70} + x_{14,71} + x_{14,72} + x_{14,73} + x_{14,74} + x_{14,75} + x_{14,76} = 1\\
 & x_{15,1} + x_{15,2} + x_{15,3} + x_{15,4} + x_{15,5} + x_{15,6} + x_{15,7} + x_{15,8} + x_{15,9} + x_{15,10} + x_{15,11} + x_{15,12} + x_{15,13} + x_{15,14} + x_{15,15} + x_{15,16} + x_{15,17} + x_{15,18} + x_{15,19} + x_{15,20} + x_{15,21} + x_{15,22} + x_{15,23} + x_{15,24} + x_{15,25} + x_{15,26} + x_{15,27} + x_{15,28} + x_{15,29} + x_{15,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{15,47} + x_{15,48} + x_{15,49} + x_{15,50} + x_{15,51} + x_{15,52} + x_{15,53} + x_{15,54} + x_{15,55} + x_{15,56} + x_{15,57} + x_{15,58} + x_{15,59} + x_{15,60} + x_{15,61} + x_{15,62} + x_{15,63} + x_{15,64} + x_{15,65} + x_{15,66} + x_{15,67} + x_{15,68} + x_{15,69} + x_{15,70} + x_{15,71} + x_{15,72} + x_{15,73} + x_{15,74} + x_{15,75} + x_{15,76} = 1\\
 & x_{16,1} + x_{16,2} + x_{16,3} + x_{16,4} + x_{16,5} + x_{16,6} + x_{16,7} + x_{16,8} + x_{16,9} + x_{16,10} + x_{16,11} + x_{16,12} + x_{16,13} + x_{16,14} + x_{16,15} + x_{16,16} + x_{16,17} + x_{16,18} + x_{16,19} + x_{16,20} + x_{16,21} + x_{16,22} + x_{16,23} + x_{16,24} + x_{16,25} + x_{16,26} + x_{16,27} + x_{16,28} + x_{16,29} + x_{16,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{16,47} + x_{16,48} + x_{16,49} + x_{16,50} + x_{16,51} + x_{16,52} + x_{16,53} + x_{16,54} + x_{16,55} + x_{16,56} + x_{16,57} + x_{16,58} + x_{16,59} + x_{16,60} + x_{16,61} + x_{16,62} + x_{16,63} + x_{16,64} + x_{16,65} + x_{16,66} + x_{16,67} + x_{16,68} + x_{16,69} + x_{16,70} + x_{16,71} + x_{16,72} + x_{16,73} + x_{16,74} + x_{16,75} + x_{16,76} = 1\\
 & x_{17,1} + x_{17,2} + x_{17,3} + x_{17,4} + x_{17,5} + x_{17,6} + x_{17,7} + x_{17,8} + x_{17,9} + x_{17,10} + x_{17,11} + x_{17,12} + x_{17,13} + x_{17,14} + x_{17,15} + x_{17,16} + x_{17,17} + x_{17,18} + x_{17,19} + x_{17,20} + x_{17,21} + x_{17,22} + x_{17,23} + x_{17,24} + x_{17,25} + x_{17,26} + x_{17,27} + x_{17,28} + x_{17,29} + x_{17,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{17,47} + x_{17,48} + x_{17,49} + x_{17,50} + x_{17,51} + x_{17,52} + x_{17,53} + x_{17,54} + x_{17,55} + x_{17,56} + x_{17,57} + x_{17,58} + x_{17,59} + x_{17,60} + x_{17,61} + x_{17,62} + x_{17,63} + x_{17,64} + x_{17,65} + x_{17,66} + x_{17,67} + x_{17,68} + x_{17,69} + x_{17,70} + x_{17,71} + x_{17,72} + x_{17,73} + x_{17,74} + x_{17,75} + x_{17,76} = 1\\
 & x_{18,1} + x_{18,2} + x_{18,3} + x_{18,4} + x_{18,5} + x_{18,6} + x_{18,7} + x_{18,8} + x_{18,9} + x_{18,10} + x_{18,11} + x_{18,12} + x_{18,13} + x_{18,14} + x_{18,15} + x_{18,16} + x_{18,17} + x_{18,18} + x_{18,19} + x_{18,20} + x_{18,21} + x_{18,22} + x_{18,23} + x_{18,24} + x_{18,25} + x_{18,26} + x_{18,27} + x_{18,28} + x_{18,29} + x_{18,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{18,47} + x_{18,48} + x_{18,49} + x_{18,50} + x_{18,51} + x_{18,52} + x_{18,53} + x_{18,54} + x_{18,55} + x_{18,56} + x_{18,57} + x_{18,58} + x_{18,59} + x_{18,60} + x_{18,61} + x_{18,62} + x_{18,63} + x_{18,64} + x_{18,65} + x_{18,66} + x_{18,67} + x_{18,68} + x_{18,69} + x_{18,70} + x_{18,71} + x_{18,72} + x_{18,73} + x_{18,74} + x_{18,75} + x_{18,76} = 1\\
 & x_{19,1} + x_{19,2} + x_{19,3} + x_{19,4} + x_{19,5} + x_{19,6} + x_{19,7} + x_{19,8} + x_{19,9} + x_{19,10} + x_{19,11} + x_{19,12} + x_{19,13} + x_{19,14} + x_{19,15} + x_{19,16} + x_{19,17} + x_{19,18} + x_{19,19} + x_{19,20} + x_{19,21} + x_{19,22} + x_{19,23} + x_{19,24} + x_{19,25} + x_{19,26} + x_{19,27} + x_{19,28} + x_{19,29} + x_{19,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{19,47} + x_{19,48} + x_{19,49} + x_{19,50} + x_{19,51} + x_{19,52} + x_{19,53} + x_{19,54} + x_{19,55} + x_{19,56} + x_{19,57} + x_{19,58} + x_{19,59} + x_{19,60} + x_{19,61} + x_{19,62} + x_{19,63} + x_{19,64} + x_{19,65} + x_{19,66} + x_{19,67} + x_{19,68} + x_{19,69} + x_{19,70} + x_{19,71} + x_{19,72} + x_{19,73} + x_{19,74} + x_{19,75} + x_{19,76} = 1\\
 & x_{20,1} + x_{20,2} + x_{20,3} + x_{20,4} + x_{20,5} + x_{20,6} + x_{20,7} + x_{20,8} + x_{20,9} + x_{20,10} + x_{20,11} + x_{20,12} + x_{20,13} + x_{20,14} + x_{20,15} + x_{20,16} + x_{20,17} + x_{20,18} + x_{20,19} + x_{20,20} + x_{20,21} + x_{20,22} + x_{20,23} + x_{20,24} + x_{20,25} + x_{20,26} + x_{20,27} + x_{20,28} + x_{20,29} + x_{20,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{20,47} + x_{20,48} + x_{20,49} + x_{20,50} + x_{20,51} + x_{20,52} + x_{20,53} + x_{20,54} + x_{20,55} + x_{20,56} + x_{20,57} + x_{20,58} + x_{20,59} + x_{20,60} + x_{20,61} + x_{20,62} + x_{20,63} + x_{20,64} + x_{20,65} + x_{20,66} + x_{20,67} + x_{20,68} + x_{20,69} + x_{20,70} + x_{20,71} + x_{20,72} + x_{20,73} + x_{20,74} + x_{20,75} + x_{20,76} = 1\\
 & x_{21,1} + x_{21,2} + x_{21,3} + x_{21,4} + x_{21,5} + x_{21,6} + x_{21,7} + x_{21,8} + x_{21,9} + x_{21,10} + x_{21,11} + x_{21,12} + x_{21,13} + x_{21,14} + x_{21,15} + x_{21,16} + x_{21,17} + x_{21,18} + x_{21,19} + x_{21,20} + x_{21,21} + x_{21,22} + x_{21,23} + x_{21,24} + x_{21,25} + x_{21,26} + x_{21,27} + x_{21,28} + x_{21,29} + x_{21,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{21,47} + x_{21,48} + x_{21,49} + x_{21,50} + x_{21,51} + x_{21,52} + x_{21,53} + x_{21,54} + x_{21,55} + x_{21,56} + x_{21,57} + x_{21,58} + x_{21,59} + x_{21,60} + x_{21,61} + x_{21,62} + x_{21,63} + x_{21,64} + x_{21,65} + x_{21,66} + x_{21,67} + x_{21,68} + x_{21,69} + x_{21,70} + x_{21,71} + x_{21,72} + x_{21,73} + x_{21,74} + x_{21,75} + x_{21,76} = 1\\
 & x_{22,1} + x_{22,2} + x_{22,3} + x_{22,4} + x_{22,5} + x_{22,6} + x_{22,7} + x_{22,8} + x_{22,9} + x_{22,10} + x_{22,11} + x_{22,12} + x_{22,13} + x_{22,14} + x_{22,15} + x_{22,16} + x_{22,17} + x_{22,18} + x_{22,19} + x_{22,20} + x_{22,21} + x_{22,22} + x_{22,23} + x_{22,24} + x_{22,25} + x_{22,26} + x_{22,27} + x_{22,28} + x_{22,29} + x_{22,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{22,47} + x_{22,48} + x_{22,49} + x_{22,50} + x_{22,51} + x_{22,52} + x_{22,53} + x_{22,54} + x_{22,55} + x_{22,56} + x_{22,57} + x_{22,58} + x_{22,59} + x_{22,60} + x_{22,61} + x_{22,62} + x_{22,63} + x_{22,64} + x_{22,65} + x_{22,66} + x_{22,67} + x_{22,68} + x_{22,69} + x_{22,70} + x_{22,71} + x_{22,72} + x_{22,73} + x_{22,74} + x_{22,75} + x_{22,76} = 1\\
 & x_{23,1} + x_{23,2} + x_{23,3} + x_{23,4} + x_{23,5} + x_{23,6} + x_{23,7} + x_{23,8} + x_{23,9} + x_{23,10} + x_{23,11} + x_{23,12} + x_{23,13} + x_{23,14} + x_{23,15} + x_{23,16} + x_{23,17} + x_{23,18} + x_{23,19} + x_{23,20} + x_{23,21} + x_{23,22} + x_{23,23} + x_{23,24} + x_{23,25} + x_{23,26} + x_{23,27} + x_{23,28} + x_{23,29} + x_{23,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{23,47} + x_{23,48} + x_{23,49} + x_{23,50} + x_{23,51} + x_{23,52} + x_{23,53} + x_{23,54} + x_{23,55} + x_{23,56} + x_{23,57} + x_{23,58} + x_{23,59} + x_{23,60} + x_{23,61} + x_{23,62} + x_{23,63} + x_{23,64} + x_{23,65} + x_{23,66} + x_{23,67} + x_{23,68} + x_{23,69} + x_{23,70} + x_{23,71} + x_{23,72} + x_{23,73} + x_{23,74} + x_{23,75} + x_{23,76} = 1\\
 & x_{24,1} + x_{24,2} + x_{24,3} + x_{24,4} + x_{24,5} + x_{24,6} + x_{24,7} + x_{24,8} + x_{24,9} + x_{24,10} + x_{24,11} + x_{24,12} + x_{24,13} + x_{24,14} + x_{24,15} + x_{24,16} + x_{24,17} + x_{24,18} + x_{24,19} + x_{24,20} + x_{24,21} + x_{24,22} + x_{24,23} + x_{24,24} + x_{24,25} + x_{24,26} + x_{24,27} + x_{24,28} + x_{24,29} + x_{24,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{24,47} + x_{24,48} + x_{24,49} + x_{24,50} + x_{24,51} + x_{24,52} + x_{24,53} + x_{24,54} + x_{24,55} + x_{24,56} + x_{24,57} + x_{24,58} + x_{24,59} + x_{24,60} + x_{24,61} + x_{24,62} + x_{24,63} + x_{24,64} + x_{24,65} + x_{24,66} + x_{24,67} + x_{24,68} + x_{24,69} + x_{24,70} + x_{24,71} + x_{24,72} + x_{24,73} + x_{24,74} + x_{24,75} + x_{24,76} = 1\\
 & x_{25,1} + x_{25,2} + x_{25,3} + x_{25,4} + x_{25,5} + x_{25,6} + x_{25,7} + x_{25,8} + x_{25,9} + x_{25,10} + x_{25,11} + x_{25,12} + x_{25,13} + x_{25,14} + x_{25,15} + x_{25,16} + x_{25,17} + x_{25,18} + x_{25,19} + x_{25,20} + x_{25,21} + x_{25,22} + x_{25,23} + x_{25,24} + x_{25,25} + x_{25,26} + x_{25,27} + x_{25,28} + x_{25,29} + x_{25,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{25,47} + x_{25,48} + x_{25,49} + x_{25,50} + x_{25,51} + x_{25,52} + x_{25,53} + x_{25,54} + x_{25,55} + x_{25,56} + x_{25,57} + x_{25,58} + x_{25,59} + x_{25,60} + x_{25,61} + x_{25,62} + x_{25,63} + x_{25,64} + x_{25,65} + x_{25,66} + x_{25,67} + x_{25,68} + x_{25,69} + x_{25,70} + x_{25,71} + x_{25,72} + x_{25,73} + x_{25,74} + x_{25,75} + x_{25,76} = 1\\
 & x_{26,1} + x_{26,2} + x_{26,3} + x_{26,4} + x_{26,5} + x_{26,6} + x_{26,7} + x_{26,8} + x_{26,9} + x_{26,10} + x_{26,11} + x_{26,12} + x_{26,13} + x_{26,14} + x_{26,15} + x_{26,16} + x_{26,17} + x_{26,18} + x_{26,19} + x_{26,20} + x_{26,21} + x_{26,22} + x_{26,23} + x_{26,24} + x_{26,25} + x_{26,26} + x_{26,27} + x_{26,28} + x_{26,29} + x_{26,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{26,47} + x_{26,48} + x_{26,49} + x_{26,50} + x_{26,51} + x_{26,52} + x_{26,53} + x_{26,54} + x_{26,55} + x_{26,56} + x_{26,57} + x_{26,58} + x_{26,59} + x_{26,60} + x_{26,61} + x_{26,62} + x_{26,63} + x_{26,64} + x_{26,65} + x_{26,66} + x_{26,67} + x_{26,68} + x_{26,69} + x_{26,70} + x_{26,71} + x_{26,72} + x_{26,73} + x_{26,74} + x_{26,75} + x_{26,76} = 1\\
 & x_{27,1} + x_{27,2} + x_{27,3} + x_{27,4} + x_{27,5} + x_{27,6} + x_{27,7} + x_{27,8} + x_{27,9} + x_{27,10} + x_{27,11} + x_{27,12} + x_{27,13} + x_{27,14} + x_{27,15} + x_{27,16} + x_{27,17} + x_{27,18} + x_{27,19} + x_{27,20} + x_{27,21} + x_{27,22} + x_{27,23} + x_{27,24} + x_{27,25} + x_{27,26} + x_{27,27} + x_{27,28} + x_{27,29} + x_{27,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{27,47} + x_{27,48} + x_{27,49} + x_{27,50} + x_{27,51} + x_{27,52} + x_{27,53} + x_{27,54} + x_{27,55} + x_{27,56} + x_{27,57} + x_{27,58} + x_{27,59} + x_{27,60} + x_{27,61} + x_{27,62} + x_{27,63} + x_{27,64} + x_{27,65} + x_{27,66} + x_{27,67} + x_{27,68} + x_{27,69} + x_{27,70} + x_{27,71} + x_{27,72} + x_{27,73} + x_{27,74} + x_{27,75} + x_{27,76} = 1\\
 & x_{28,1} + x_{28,2} + x_{28,3} + x_{28,4} + x_{28,5} + x_{28,6} + x_{28,7} + x_{28,8} + x_{28,9} + x_{28,10} + x_{28,11} + x_{28,12} + x_{28,13} + x_{28,14} + x_{28,15} + x_{28,16} + x_{28,17} + x_{28,18} + x_{28,19} + x_{28,20} + x_{28,21} + x_{28,22} + x_{28,23} + x_{28,24} + x_{28,25} + x_{28,26} + x_{28,27} + x_{28,28} + x_{28,29} + x_{28,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{28,47} + x_{28,48} + x_{28,49} + x_{28,50} + x_{28,51} + x_{28,52} + x_{28,53} + x_{28,54} + x_{28,55} + x_{28,56} + x_{28,57} + x_{28,58} + x_{28,59} + x_{28,60} + x_{28,61} + x_{28,62} + x_{28,63} + x_{28,64} + x_{28,65} + x_{28,66} + x_{28,67} + x_{28,68} + x_{28,69} + x_{28,70} + x_{28,71} + x_{28,72} + x_{28,73} + x_{28,74} + x_{28,75} + x_{28,76} = 1\\
 & x_{29,1} + x_{29,2} + x_{29,3} + x_{29,4} + x_{29,5} + x_{29,6} + x_{29,7} + x_{29,8} + x_{29,9} + x_{29,10} + x_{29,11} + x_{29,12} + x_{29,13} + x_{29,14} + x_{29,15} + x_{29,16} + x_{29,17} + x_{29,18} + x_{29,19} + x_{29,20} + x_{29,21} + x_{29,22} + x_{29,23} + x_{29,24} + x_{29,25} + x_{29,26} + x_{29,27} + x_{29,28} + x_{29,29} + x_{29,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{29,47} + x_{29,48} + x_{29,49} + x_{29,50} + x_{29,51} + x_{29,52} + x_{29,53} + x_{29,54} + x_{29,55} + x_{29,56} + x_{29,57} + x_{29,58} + x_{29,59} + x_{29,60} + x_{29,61} + x_{29,62} + x_{29,63} + x_{29,64} + x_{29,65} + x_{29,66} + x_{29,67} + x_{29,68} + x_{29,69} + x_{29,70} + x_{29,71} + x_{29,72} + x_{29,73} + x_{29,74} + x_{29,75} + x_{29,76} = 1\\
 & x_{30,1} + x_{30,2} + x_{30,3} + x_{30,4} + x_{30,5} + x_{30,6} + x_{30,7} + x_{30,8} + x_{30,9} + x_{30,10} + x_{30,11} + x_{30,12} + x_{30,13} + x_{30,14} + x_{30,15} + x_{30,16} + x_{30,17} + x_{30,18} + x_{30,19} + x_{30,20} + x_{30,21} + x_{30,22} + x_{30,23} + x_{30,24} + x_{30,25} + x_{30,26} + x_{30,27} + x_{30,28} + x_{30,29} + x_{30,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{30,47} + x_{30,48} + x_{30,49} + x_{30,50} + x_{30,51} + x_{30,52} + x_{30,53} + x_{30,54} + x_{30,55} + x_{30,56} + x_{30,57} + x_{30,58} + x_{30,59} + x_{30,60} + x_{30,61} + x_{30,62} + x_{30,63} + x_{30,64} + x_{30,65} + x_{30,66} + x_{30,67} + x_{30,68} + x_{30,69} + x_{30,70} + x_{30,71} + x_{30,72} + x_{30,73} + x_{30,74} + x_{30,75} + x_{30,76} = 1\\
 & x_{31,1} + x_{31,2} + x_{31,3} + x_{31,4} + x_{31,5} + x_{31,6} + x_{31,7} + x_{31,8} + x_{31,9} + x_{31,10} + x_{31,11} + x_{31,12} + x_{31,13} + x_{31,14} + x_{31,15} + x_{31,16} + x_{31,17} + x_{31,18} + x_{31,19} + x_{31,20} + x_{31,21} + x_{31,22} + x_{31,23} + x_{31,24} + x_{31,25} + x_{31,26} + x_{31,27} + x_{31,28} + x_{31,29} + x_{31,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{31,47} + x_{31,48} + x_{31,49} + x_{31,50} + x_{31,51} + x_{31,52} + x_{31,53} + x_{31,54} + x_{31,55} + x_{31,56} + x_{31,57} + x_{31,58} + x_{31,59} + x_{31,60} + x_{31,61} + x_{31,62} + x_{31,63} + x_{31,64} + x_{31,65} + x_{31,66} + x_{31,67} + x_{31,68} + x_{31,69} + x_{31,70} + x_{31,71} + x_{31,72} + x_{31,73} + x_{31,74} + x_{31,75} + x_{31,76} = 1\\
 & x_{32,1} + x_{32,2} + x_{32,3} + x_{32,4} + x_{32,5} + x_{32,6} + x_{32,7} + x_{32,8} + x_{32,9} + x_{32,10} + x_{32,11} + x_{32,12} + x_{32,13} + x_{32,14} + x_{32,15} + x_{32,16} + x_{32,17} + x_{32,18} + x_{32,19} + x_{32,20} + x_{32,21} + x_{32,22} + x_{32,23} + x_{32,24} + x_{32,25} + x_{32,26} + x_{32,27} + x_{32,28} + x_{32,29} + x_{32,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{32,47} + x_{32,48} + x_{32,49} + x_{32,50} + x_{32,51} + x_{32,52} + x_{32,53} + x_{32,54} + x_{32,55} + x_{32,56} + x_{32,57} + x_{32,58} + x_{32,59} + x_{32,60} + x_{32,61} + x_{32,62} + x_{32,63} + x_{32,64} + x_{32,65} + x_{32,66} + x_{32,67} + x_{32,68} + x_{32,69} + x_{32,70} + x_{32,71} + x_{32,72} + x_{32,73} + x_{32,74} + x_{32,75} + x_{32,76} = 1\\
 & x_{33,1} + x_{33,2} + x_{33,3} + x_{33,4} + x_{33,5} + x_{33,6} + x_{33,7} + x_{33,8} + x_{33,9} + x_{33,10} + x_{33,11} + x_{33,12} + x_{33,13} + x_{33,14} + x_{33,15} + x_{33,16} + x_{33,17} + x_{33,18} + x_{33,19} + x_{33,20} + x_{33,21} + x_{33,22} + x_{33,23} + x_{33,24} + x_{33,25} + x_{33,26} + x_{33,27} + x_{33,28} + x_{33,29} + x_{33,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{33,47} + x_{33,48} + x_{33,49} + x_{33,50} + x_{33,51} + x_{33,52} + x_{33,53} + x_{33,54} + x_{33,55} + x_{33,56} + x_{33,57} + x_{33,58} + x_{33,59} + x_{33,60} + x_{33,61} + x_{33,62} + x_{33,63} + x_{33,64} + x_{33,65} + x_{33,66} + x_{33,67} + x_{33,68} + x_{33,69} + x_{33,70} + x_{33,71} + x_{33,72} + x_{33,73} + x_{33,74} + x_{33,75} + x_{33,76} = 1\\
 & x_{34,1} + x_{34,2} + x_{34,3} + x_{34,4} + x_{34,5} + x_{34,6} + x_{34,7} + x_{34,8} + x_{34,9} + x_{34,10} + x_{34,11} + x_{34,12} + x_{34,13} + x_{34,14} + x_{34,15} + x_{34,16} + x_{34,17} + x_{34,18} + x_{34,19} + x_{34,20} + x_{34,21} + x_{34,22} + x_{34,23} + x_{34,24} + x_{34,25} + x_{34,26} + x_{34,27} + x_{34,28} + x_{34,29} + x_{34,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{34,47} + x_{34,48} + x_{34,49} + x_{34,50} + x_{34,51} + x_{34,52} + x_{34,53} + x_{34,54} + x_{34,55} + x_{34,56} + x_{34,57} + x_{34,58} + x_{34,59} + x_{34,60} + x_{34,61} + x_{34,62} + x_{34,63} + x_{34,64} + x_{34,65} + x_{34,66} + x_{34,67} + x_{34,68} + x_{34,69} + x_{34,70} + x_{34,71} + x_{34,72} + x_{34,73} + x_{34,74} + x_{34,75} + x_{34,76} = 1\\
 & x_{35,1} + x_{35,2} + x_{35,3} + x_{35,4} + x_{35,5} + x_{35,6} + x_{35,7} + x_{35,8} + x_{35,9} + x_{35,10} + x_{35,11} + x_{35,12} + x_{35,13} + x_{35,14} + x_{35,15} + x_{35,16} + x_{35,17} + x_{35,18} + x_{35,19} + x_{35,20} + x_{35,21} + x_{35,22} + x_{35,23} + x_{35,24} + x_{35,25} + x_{35,26} + x_{35,27} + x_{35,28} + x_{35,29} + x_{35,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{35,47} + x_{35,48} + x_{35,49} + x_{35,50} + x_{35,51} + x_{35,52} + x_{35,53} + x_{35,54} + x_{35,55} + x_{35,56} + x_{35,57} + x_{35,58} + x_{35,59} + x_{35,60} + x_{35,61} + x_{35,62} + x_{35,63} + x_{35,64} + x_{35,65} + x_{35,66} + x_{35,67} + x_{35,68} + x_{35,69} + x_{35,70} + x_{35,71} + x_{35,72} + x_{35,73} + x_{35,74} + x_{35,75} + x_{35,76} = 1\\
 & x_{36,1} + x_{36,2} + x_{36,3} + x_{36,4} + x_{36,5} + x_{36,6} + x_{36,7} + x_{36,8} + x_{36,9} + x_{36,10} + x_{36,11} + x_{36,12} + x_{36,13} + x_{36,14} + x_{36,15} + x_{36,16} + x_{36,17} + x_{36,18} + x_{36,19} + x_{36,20} + x_{36,21} + x_{36,22} + x_{36,23} + x_{36,24} + x_{36,25} + x_{36,26} + x_{36,27} + x_{36,28} + x_{36,29} + x_{36,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{36,47} + x_{36,48} + x_{36,49} + x_{36,50} + x_{36,51} + x_{36,52} + x_{36,53} + x_{36,54} + x_{36,55} + x_{36,56} + x_{36,57} + x_{36,58} + x_{36,59} + x_{36,60} + x_{36,61} + x_{36,62} + x_{36,63} + x_{36,64} + x_{36,65} + x_{36,66} + x_{36,67} + x_{36,68} + x_{36,69} + x_{36,70} + x_{36,71} + x_{36,72} + x_{36,73} + x_{36,74} + x_{36,75} + x_{36,76} = 1\\
 & x_{37,1} + x_{37,2} + x_{37,3} + x_{37,4} + x_{37,5} + x_{37,6} + x_{37,7} + x_{37,8} + x_{37,9} + x_{37,10} + x_{37,11} + x_{37,12} + x_{37,13} + x_{37,14} + x_{37,15} + x_{37,16} + x_{37,17} + x_{37,18} + x_{37,19} + x_{37,20} + x_{37,21} + x_{37,22} + x_{37,23} + x_{37,24} + x_{37,25} + x_{37,26} + x_{37,27} + x_{37,28} + x_{37,29} + x_{37,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{37,47} + x_{37,48} + x_{37,49} + x_{37,50} + x_{37,51} + x_{37,52} + x_{37,53} + x_{37,54} + x_{37,55} + x_{37,56} + x_{37,57} + x_{37,58} + x_{37,59} + x_{37,60} + x_{37,61} + x_{37,62} + x_{37,63} + x_{37,64} + x_{37,65} + x_{37,66} + x_{37,67} + x_{37,68} + x_{37,69} + x_{37,70} + x_{37,71} + x_{37,72} + x_{37,73} + x_{37,74} + x_{37,75} + x_{37,76} = 1\\
 & x_{38,1} + x_{38,2} + x_{38,3} + x_{38,4} + x_{38,5} + x_{38,6} + x_{38,7} + x_{38,8} + x_{38,9} + x_{38,10} + x_{38,11} + x_{38,12} + x_{38,13} + x_{38,14} + x_{38,15} + x_{38,16} + x_{38,17} + x_{38,18} + x_{38,19} + x_{38,20} + x_{38,21} + x_{38,22} + x_{38,23} + x_{38,24} + x_{38,25} + x_{38,26} + x_{38,27} + x_{38,28} + x_{38,29} + x_{38,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{38,47} + x_{38,48} + x_{38,49} + x_{38,50} + x_{38,51} + x_{38,52} + x_{38,53} + x_{38,54} + x_{38,55} + x_{38,56} + x_{38,57} + x_{38,58} + x_{38,59} + x_{38,60} + x_{38,61} + x_{38,62} + x_{38,63} + x_{38,64} + x_{38,65} + x_{38,66} + x_{38,67} + x_{38,68} + x_{38,69} + x_{38,70} + x_{38,71} + x_{38,72} + x_{38,73} + x_{38,74} + x_{38,75} + x_{38,76} = 1\\
 & x_{39,1} + x_{39,2} + x_{39,3} + x_{39,4} + x_{39,5} + x_{39,6} + x_{39,7} + x_{39,8} + x_{39,9} + x_{39,10} + x_{39,11} + x_{39,12} + x_{39,13} + x_{39,14} + x_{39,15} + x_{39,16} + x_{39,17} + x_{39,18} + x_{39,19} + x_{39,20} + x_{39,21} + x_{39,22} + x_{39,23} + x_{39,24} + x_{39,25} + x_{39,26} + x_{39,27} + x_{39,28} + x_{39,29} + x_{39,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{39,47} + x_{39,48} + x_{39,49} + x_{39,50} + x_{39,51} + x_{39,52} + x_{39,53} + x_{39,54} + x_{39,55} + x_{39,56} + x_{39,57} + x_{39,58} + x_{39,59} + x_{39,60} + x_{39,61} + x_{39,62} + x_{39,63} + x_{39,64} + x_{39,65} + x_{39,66} + x_{39,67} + x_{39,68} + x_{39,69} + x_{39,70} + x_{39,71} + x_{39,72} + x_{39,73} + x_{39,74} + x_{39,75} + x_{39,76} = 1\\
 & x_{40,1} + x_{40,2} + x_{40,3} + x_{40,4} + x_{40,5} + x_{40,6} + x_{40,7} + x_{40,8} + x_{40,9} + x_{40,10} + x_{40,11} + x_{40,12} + x_{40,13} + x_{40,14} + x_{40,15} + x_{40,16} + x_{40,17} + x_{40,18} + x_{40,19} + x_{40,20} + x_{40,21} + x_{40,22} + x_{40,23} + x_{40,24} + x_{40,25} + x_{40,26} + x_{40,27} + x_{40,28} + x_{40,29} + x_{40,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{40,47} + x_{40,48} + x_{40,49} + x_{40,50} + x_{40,51} + x_{40,52} + x_{40,53} + x_{40,54} + x_{40,55} + x_{40,56} + x_{40,57} + x_{40,58} + x_{40,59} + x_{40,60} + x_{40,61} + x_{40,62} + x_{40,63} + x_{40,64} + x_{40,65} + x_{40,66} + x_{40,67} + x_{40,68} + x_{40,69} + x_{40,70} + x_{40,71} + x_{40,72} + x_{40,73} + x_{40,74} + x_{40,75} + x_{40,76} = 1\\
 & x_{41,1} + x_{41,2} + x_{41,3} + x_{41,4} + x_{41,5} + x_{41,6} + x_{41,7} + x_{41,8} + x_{41,9} + x_{41,10} + x_{41,11} + x_{41,12} + x_{41,13} + x_{41,14} + x_{41,15} + x_{41,16} + x_{41,17} + x_{41,18} + x_{41,19} + x_{41,20} + x_{41,21} + x_{41,22} + x_{41,23} + x_{41,24} + x_{41,25} + x_{41,26} + x_{41,27} + x_{41,28} + x_{41,29} + x_{41,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{41,47} + x_{41,48} + x_{41,49} + x_{41,50} + x_{41,51} + x_{41,52} + x_{41,53} + x_{41,54} + x_{41,55} + x_{41,56} + x_{41,57} + x_{41,58} + x_{41,59} + x_{41,60} + x_{41,61} + x_{41,62} + x_{41,63} + x_{41,64} + x_{41,65} + x_{41,66} + x_{41,67} + x_{41,68} + x_{41,69} + x_{41,70} + x_{41,71} + x_{41,72} + x_{41,73} + x_{41,74} + x_{41,75} + x_{41,76} = 1\\
 & x_{42,1} + x_{42,2} + x_{42,3} + x_{42,4} + x_{42,5} + x_{42,6} + x_{42,7} + x_{42,8} + x_{42,9} + x_{42,10} + x_{42,11} + x_{42,12} + x_{42,13} + x_{42,14} + x_{42,15} + x_{42,16} + x_{42,17} + x_{42,18} + x_{42,19} + x_{42,20} + x_{42,21} + x_{42,22} + x_{42,23} + x_{42,24} + x_{42,25} + x_{42,26} + x_{42,27} + x_{42,28} + x_{42,29} + x_{42,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{42,47} + x_{42,48} + x_{42,49} + x_{42,50} + x_{42,51} + x_{42,52} + x_{42,53} + x_{42,54} + x_{42,55} + x_{42,56} + x_{42,57} + x_{42,58} + x_{42,59} + x_{42,60} + x_{42,61} + x_{42,62} + x_{42,63} + x_{42,64} + x_{42,65} + x_{42,66} + x_{42,67} + x_{42,68} + x_{42,69} + x_{42,70} + x_{42,71} + x_{42,72} + x_{42,73} + x_{42,74} + x_{42,75} + x_{42,76} = 1\\
 & x_{43,1} + x_{43,2} + x_{43,3} + x_{43,4} + x_{43,5} + x_{43,6} + x_{43,7} + x_{43,8} + x_{43,9} + x_{43,10} + x_{43,11} + x_{43,12} + x_{43,13} + x_{43,14} + x_{43,15} + x_{43,16} + x_{43,17} + x_{43,18} + x_{43,19} + x_{43,20} + x_{43,21} + x_{43,22} + x_{43,23} + x_{43,24} + x_{43,25} + x_{43,26} + x_{43,27} + x_{43,28} + x_{43,29} + x_{43,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{43,47} + x_{43,48} + x_{43,49} + x_{43,50} + x_{43,51} + x_{43,52} + x_{43,53} + x_{43,54} + x_{43,55} + x_{43,56} + x_{43,57} + x_{43,58} + x_{43,59} + x_{43,60} + x_{43,61} + x_{43,62} + x_{43,63} + x_{43,64} + x_{43,65} + x_{43,66} + x_{43,67} + x_{43,68} + x_{43,69} + x_{43,70} + x_{43,71} + x_{43,72} + x_{43,73} + x_{43,74} + x_{43,75} + x_{43,76} = 1\\
 & x_{44,1} + x_{44,2} + x_{44,3} + x_{44,4} + x_{44,5} + x_{44,6} + x_{44,7} + x_{44,8} + x_{44,9} + x_{44,10} + x_{44,11} + x_{44,12} + x_{44,13} + x_{44,14} + x_{44,15} + x_{44,16} + x_{44,17} + x_{44,18} + x_{44,19} + x_{44,20} + x_{44,21} + x_{44,22} + x_{44,23} + x_{44,24} + x_{44,25} + x_{44,26} + x_{44,27} + x_{44,28} + x_{44,29} + x_{44,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{44,47} + x_{44,48} + x_{44,49} + x_{44,50} + x_{44,51} + x_{44,52} + x_{44,53} + x_{44,54} + x_{44,55} + x_{44,56} + x_{44,57} + x_{44,58} + x_{44,59} + x_{44,60} + x_{44,61} + x_{44,62} + x_{44,63} + x_{44,64} + x_{44,65} + x_{44,66} + x_{44,67} + x_{44,68} + x_{44,69} + x_{44,70} + x_{44,71} + x_{44,72} + x_{44,73} + x_{44,74} + x_{44,75} + x_{44,76} = 1\\
 & x_{45,1} + x_{45,2} + x_{45,3} + x_{45,4} + x_{45,5} + x_{45,6} + x_{45,7} + x_{45,8} + x_{45,9} + x_{45,10} + x_{45,11} + x_{45,12} + x_{45,13} + x_{45,14} + x_{45,15} + x_{45,16} + x_{45,17} + x_{45,18} + x_{45,19} + x_{45,20} + x_{45,21} + x_{45,22} + x_{45,23} + x_{45,24} + x_{45,25} + x_{45,26} + x_{45,27} + x_{45,28} + x_{45,29} + x_{45,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{45,47} + x_{45,48} + x_{45,49} + x_{45,50} + x_{45,51} + x_{45,52} + x_{45,53} + x_{45,54} + x_{45,55} + x_{45,56} + x_{45,57} + x_{45,58} + x_{45,59} + x_{45,60} + x_{45,61} + x_{45,62} + x_{45,63} + x_{45,64} + x_{45,65} + x_{45,66} + x_{45,67} + x_{45,68} + x_{45,69} + x_{45,70} + x_{45,71} + x_{45,72} + x_{45,73} + x_{45,74} + x_{45,75} + x_{45,76} = 1\\
 & x_{46,1} + x_{46,2} + x_{46,3} + x_{46,4} + x_{46,5} + x_{46,6} + x_{46,7} + x_{46,8} + x_{46,9} + x_{46,10} + x_{46,11} + x_{46,12} + x_{46,13} + x_{46,14} + x_{46,15} + x_{46,16} + x_{46,17} + x_{46,18} + x_{46,19} + x_{46,20} + x_{46,21} + x_{46,22} + x_{46,23} + x_{46,24} + x_{46,25} + x_{46,26} + x_{46,27} + x_{46,28} + x_{46,29} + x_{46,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{46,47} + x_{46,48} + x_{46,49} + x_{46,50} + x_{46,51} + x_{46,52} + x_{46,53} + x_{46,54} + x_{46,55} + x_{46,56} + x_{46,57} + x_{46,58} + x_{46,59} + x_{46,60} + x_{46,61} + x_{46,62} + x_{46,63} + x_{46,64} + x_{46,65} + x_{46,66} + x_{46,67} + x_{46,68} + x_{46,69} + x_{46,70} + x_{46,71} + x_{46,72} + x_{46,73} + x_{46,74} + x_{46,75} + x_{46,76} = 1\\
 & x_{47,1} + x_{47,2} + x_{47,3} + x_{47,4} + x_{47,5} + x_{47,6} + x_{47,7} + x_{47,8} + x_{47,9} + x_{47,10} + x_{47,11} + x_{47,12} + x_{47,13} + x_{47,14} + x_{47,15} + x_{47,16} + x_{47,17} + x_{47,18} + x_{47,19} + x_{47,20} + x_{47,21} + x_{47,22} + x_{47,23} + x_{47,24} + x_{47,25} + x_{47,26} + x_{47,27} + x_{47,28} + x_{47,29} + x_{47,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{47,47} + x_{47,48} + x_{47,49} + x_{47,50} + x_{47,51} + x_{47,52} + x_{47,53} + x_{47,54} + x_{47,55} + x_{47,56} + x_{47,57} + x_{47,58} + x_{47,59} + x_{47,60} + x_{47,61} + x_{47,62} + x_{47,63} + x_{47,64} + x_{47,65} + x_{47,66} + x_{47,67} + x_{47,68} + x_{47,69} + x_{47,70} + x_{47,71} + x_{47,72} + x_{47,73} + x_{47,74} + x_{47,75} + x_{47,76} = 1\\
 & x_{48,1} + x_{48,2} + x_{48,3} + x_{48,4} + x_{48,5} + x_{48,6} + x_{48,7} + x_{48,8} + x_{48,9} + x_{48,10} + x_{48,11} + x_{48,12} + x_{48,13} + x_{48,14} + x_{48,15} + x_{48,16} + x_{48,17} + x_{48,18} + x_{48,19} + x_{48,20} + x_{48,21} + x_{48,22} + x_{48,23} + x_{48,24} + x_{48,25} + x_{48,26} + x_{48,27} + x_{48,28} + x_{48,29} + x_{48,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{48,47} + x_{48,48} + x_{48,49} + x_{48,50} + x_{48,51} + x_{48,52} + x_{48,53} + x_{48,54} + x_{48,55} + x_{48,56} + x_{48,57} + x_{48,58} + x_{48,59} + x_{48,60} + x_{48,61} + x_{48,62} + x_{48,63} + x_{48,64} + x_{48,65} + x_{48,66} + x_{48,67} + x_{48,68} + x_{48,69} + x_{48,70} + x_{48,71} + x_{48,72} + x_{48,73} + x_{48,74} + x_{48,75} + x_{48,76} = 1\\
 & x_{49,1} + x_{49,2} + x_{49,3} + x_{49,4} + x_{49,5} + x_{49,6} + x_{49,7} + x_{49,8} + x_{49,9} + x_{49,10} + x_{49,11} + x_{49,12} + x_{49,13} + x_{49,14} + x_{49,15} + x_{49,16} + x_{49,17} + x_{49,18} + x_{49,19} + x_{49,20} + x_{49,21} + x_{49,22} + x_{49,23} + x_{49,24} + x_{49,25} + x_{49,26} + x_{49,27} + x_{49,28} + x_{49,29} + x_{49,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{49,47} + x_{49,48} + x_{49,49} + x_{49,50} + x_{49,51} + x_{49,52} + x_{49,53} + x_{49,54} + x_{49,55} + x_{49,56} + x_{49,57} + x_{49,58} + x_{49,59} + x_{49,60} + x_{49,61} + x_{49,62} + x_{49,63} + x_{49,64} + x_{49,65} + x_{49,66} + x_{49,67} + x_{49,68} + x_{49,69} + x_{49,70} + x_{49,71} + x_{49,72} + x_{49,73} + x_{49,74} + x_{49,75} + x_{49,76} = 1\\
 & x_{50,1} + x_{50,2} + x_{50,3} + x_{50,4} + x_{50,5} + x_{50,6} + x_{50,7} + x_{50,8} + x_{50,9} + x_{50,10} + x_{50,11} + x_{50,12} + x_{50,13} + x_{50,14} + x_{50,15} + x_{50,16} + x_{50,17} + x_{50,18} + x_{50,19} + x_{50,20} + x_{50,21} + x_{50,22} + x_{50,23} + x_{50,24} + x_{50,25} + x_{50,26} + x_{50,27} + x_{50,28} + x_{50,29} + x_{50,30} + [[\ldots\text{16 terms omitted}\ldots]] + x_{50,47} + x_{50,48} + x_{50,49} + x_{50,50} + x_{50,51} + x_{50,52} + x_{50,53} + x_{50,54} + x_{50,55} + x_{50,56} + x_{50,57} + x_{50,58} + x_{50,59} + x_{50,60} + x_{50,61} + x_{50,62} + x_{50,63} + x_{50,64} + x_{50,65} + x_{50,66} + x_{50,67} + x_{50,68} + x_{50,69} + x_{50,70} + x_{50,71} + x_{50,72} + x_{50,73} + x_{50,74} + x_{50,75} + x_{50,76} = 1\\
 & [[\ldots\text{183499 constraints skipped}\ldots]] \\
 & y_{27} \in \{0, 1\}\\
 & y_{28} \in \{0, 1\}\\
 & y_{29} \in \{0, 1\}\\
 & y_{30} \in \{0, 1\}\\
 & y_{31} \in \{0, 1\}\\
 & y_{32} \in \{0, 1\}\\
 & y_{33} \in \{0, 1\}\\
 & y_{34} \in \{0, 1\}\\
 & y_{35} \in \{0, 1\}\\
 & y_{36} \in \{0, 1\}\\
 & y_{37} \in \{0, 1\}\\
 & y_{38} \in \{0, 1\}\\
 & y_{39} \in \{0, 1\}\\
 & y_{40} \in \{0, 1\}\\
 & y_{41} \in \{0, 1\}\\
 & y_{42} \in \{0, 1\}\\
 & y_{43} \in \{0, 1\}\\
 & y_{44} \in \{0, 1\}\\
 & y_{45} \in \{0, 1\}\\
 & y_{46} \in \{0, 1\}\\
 & y_{47} \in \{0, 1\}\\
 & y_{48} \in \{0, 1\}\\
 & y_{49} \in \{0, 1\}\\
 & y_{50} \in \{0, 1\}\\
 & y_{51} \in \{0, 1\}\\
 & y_{52} \in \{0, 1\}\\
 & y_{53} \in \{0, 1\}\\
 & y_{54} \in \{0, 1\}\\
 & y_{55} \in \{0, 1\}\\
 & y_{56} \in \{0, 1\}\\
 & y_{57} \in \{0, 1\}\\
 & y_{58} \in \{0, 1\}\\
 & y_{59} \in \{0, 1\}\\
 & y_{60} \in \{0, 1\}\\
 & y_{61} \in \{0, 1\}\\
 & y_{62} \in \{0, 1\}\\
 & y_{63} \in \{0, 1\}\\
 & y_{64} \in \{0, 1\}\\
 & y_{65} \in \{0, 1\}\\
 & y_{66} \in \{0, 1\}\\
 & y_{67} \in \{0, 1\}\\
 & y_{68} \in \{0, 1\}\\
 & y_{69} \in \{0, 1\}\\
 & y_{70} \in \{0, 1\}\\
 & y_{71} \in \{0, 1\}\\
 & y_{72} \in \{0, 1\}\\
 & y_{73} \in \{0, 1\}\\
 & y_{74} \in \{0, 1\}\\
 & y_{75} \in \{0, 1\}\\
 & y_{76} \in \{0, 1\}\\
\end{aligned} $$

Dónde f’y es un coste fijo (que hemos puesto de 1) que imponemos por abrir cada sede (código postal) y sum(c .* x)

dónde c es la matriz ajustada de distancias entre los comerciales y las sedes (con el truco de aumentar artificialmente la distancia para los jefes) y x[i,j] es la matriz n x m que indica si la sede i se asigna a comercial j. con .* se hace la multiplicación elemento a elemento , y se suma para obtener el coste total. Al minimizar esto lo que estamos indicando es que asigne sedes a comerciales de forma que se minimice de forma global las distancias a las sedes (queremos que la gente de Almería no le asignes una sede en Huelva). Y además hemos metido las restricciones de que solo un comercial por sede y lo de que los jefes no trabajen demasiado (como mucho 10 sedes por jefe)

Pues ya solo queda dejar que JuMP haga su magia.

Show the code 

optimize!(ufl)

println("Optimal value: ", objective_value(ufl))
Running HiGHS 1.11.0 (git hash: 364c83a51e): Copyright (c) 2025 HiGHS under MIT licence terms
MIP  has 92399 rows; 91200 cols; 364496 nonzeros; 91200 integer variables (91200 binary)
Coefficient ranges:
  Matrix [1e+00, 1e+00]
  Cost   [1e+00, 5e+02]
  Bound  [1e+00, 1e+00]
  RHS    [1e+00, 1e+02]
Presolving model
92399 rows, 91200 cols, 364496 nonzeros  0s
92399 rows, 91200 cols, 364496 nonzeros  0s
Objective function is integral with scale 1000

Solving MIP model with:
   92399 rows
   91200 cols (91200 binary, 0 integer, 0 implied int., 0 continuous, 0 domain fixed)
   364496 nonzeros

Src: B => Branching; C => Central rounding; F => Feasibility pump; J => Feasibility jump;
     H => Heuristic; L => Sub-MIP; P => Empty MIP; R => Randomized rounding; Z => ZI Round;
     I => Shifting; S => Solve LP; T => Evaluate node; U => Unbounded; X => User solution;
     z => Trivial zero; l => Trivial lower; u => Trivial upper; p => Trivial point

        Nodes      |    B&B Tree     |            Objective Bounds              |  Dynamic Constraints |       Work      
Src  Proc. InQueue |  Leaves   Expl. | BestBound       BestSol              Gap |   Cuts   InLp Confl. | LpIters     Time

 J       0       0         0   0.00%   -inf            178645.113         Large        0      0      0         0     0.6s
         0       0         0   0.00%   21182.055167    178645.113        88.14%        0      0      4      1971     0.9s
 C       0       0         0   0.00%   21202.838       95153.418         77.72%        0      0      7      2198     1.0s
 L       0       0         0   0.00%   21202.838       21202.838          0.00%        0      0      7      2198     1.4s
         1       0         1 100.00%   21202.838       21202.838          0.00%        0      0      7      2228     1.4s

Solving report
  Status            Optimal
  Primal bound      21202.838
  Dual bound        21202.838
  Gap               0% (tolerance: 0.01%)
  P-D integral      0.612597059043
  Solution status   feasible
                    21202.838 (objective)
                    0 (bound viol.)
                    0 (int. viol.)
                    0 (row viol.)
  Timing            1.44 (total)
                    0.00 (presolve)
                    0.00 (solve)
                    0.00 (postsolve)
  Max sub-MIP depth 1
  Nodes             1
  Repair LPs        0 (0 feasible; 0 iterations)
  LP iterations     2228 (total)
                    0 (strong br.)
                    227 (separation)
                    30 (heuristics)
Optimal value: 21202.838
Show the code 
# value.(x) obtiene una matriz con los valores optimizados de las variables x[i,j] (pueden ser algo como 0.999999 o 1.0 por temas de redondeo).
 # x_ es una matriz booleana del mismo tamaño que x, indicando qué asignaciones cliente–sede están activas.
 # y_ devuelve true o false indicando que sedes se han habierto

x_ = value.(x) .> 1 - 1e-5
y_ = value.(y) .> 1 - 1e-5

#value.(comparacion)
value.(tot_por_indio)
value.(tot_por_jefe)
27-element Vector{Float64}:
  1.0
  8.0
  1.0
 10.0
  1.0
 10.0
  9.0
 10.0
  8.0
  1.0
  ⋮
  1.0
 10.0
  1.0
  1.0
  1.0
 10.0
 10.0
  9.0
  1.0

Pintamos, la verdad es que pintar en Julia no me agrada demasiado

Show the code 

p = Plots.scatter(
    cod_postales.centroide_longitud,
    cod_postales.centroide_latitud;
    markershape = :circle,
    markercolor = :blue,
    label = nothing,
)


mc = [(y_[j] ? :red : :white) for j in 1:m]
Plots.scatter!(
    sedes.centroide_longitud,
    sedes.centroide_latitud;
    markershape = :square,
    markercolor = mc,
    markersize = 4,
    markerstrokecolor = :red,
    markerstrokewidth = 2,
    label = nothing,
)


for i in 1:n
    for j in 1:m
        if x_[i, j] == 1
            Plots.plot!(
                [cod_postales.centroide_longitud[i], sedes.centroide_longitud[j]],
                [cod_postales.centroide_latitud[i], sedes.centroide_latitud[j]];
                color = :black,
                label = nothing,
            )
        end
    end
end

p

Ahora extraemos las asignaciones matrix x[i,j] y contamos para cada columna j cuantos códigos postales le han caído. Es decir, cuántas le han tocado a cada comercial

Show the code 
# 
x_collect= collect(x_)
#sum(collect(x_), dims=1)
sum(x_, dims = 1) 
1×76 Matrix{Int64}:
 1  8  1  10  1  10  9  10  8  1  10  1  …  8  23  55  24  13  28  24  3  10

Lo siguiente es solo comprobar que a cada cod postal solo va un único comercial.

Show the code 
sum(x_, dims = 2)
1199×1 Matrix{Int64}:
 1
 1
 1
 1
 1
 1
 1
 1
 1
 1
 ⋮
 1
 1
 1
 1
 1
 1
 1
 1
 1

Lo siguiente lo hice hace tiempo y no recuerdo muy bien lo que hace, pero el objetivo era mostrar para cada comercial, cuántos cod postales tiene que visitar. Al final salen tuplas dónde el primer elemento es el índice del comercial y el segndo el total de códigos postales que visita. Como ordenamos poniendo primero a los jefes veremos que como máximo visitan 10 códigos postales y como mínimo 1. Luego vendrán las asignaciones de los pobres indios, dónde alguno puede haber tenido fortuna y solo se le asignan 3 códigos postales. Hace falta un sindicato, claramente

Show the code 
[x for x in 1:10 if x % 2 == 0]
# comprehension lista 
asignaciones  = Dict()
#= for j in 1:m
    asignaciones("clave" => j, "valor" =>
        [index for (index, fila) 
        in enumerate(x_collect[:,j]) 
        if fila .== 1])
end  =#

asignaciones  = Dict()
for j in 1:m
    asignaciones[j] = 
    [index for (index, fila) in enumerate(x_collect[:,j]) if fila .== 1]
end 

for j in 1:m
    asignaciones[j] = 
    cod_postales[ [index for (index, fila) in enumerate(x_collect[:,j]) if fila .== 1], :cod_postal]
end 

for j in 1:m
    asignaciones[j] = 
    cod_postales[ [index for (index, fila) in enumerate(x_collect[:,j]) if fila .== 1], [:cod_postal, :provincia]]
end 


for j in 1:m
    println((j, size(asignaciones[j])[1]))
end
(1, 1)
(2, 8)
(3, 1)
(4, 10)
(5, 1)
(6, 10)
(7, 9)
(8, 10)
(9, 8)
(10, 1)
(11, 10)
(12, 1)
(13, 10)
(14, 1)
(15, 1)
(16, 1)
(17, 9)
(18, 1)
(19, 1)
(20, 10)
(21, 1)
(22, 1)
(23, 1)
(24, 10)
(25, 10)
(26, 9)
(27, 1)
(28, 70)
(29, 14)
(30, 21)
(31, 18)
(32, 4)
(33, 34)
(34, 12)
(35, 15)
(36, 16)
(37, 9)
(38, 49)
(39, 14)
(40, 6)
(41, 12)
(42, 9)
(43, 14)
(44, 24)
(45, 9)
(46, 24)
(47, 33)
(48, 49)
(49, 16)
(50, 11)
(51, 22)
(52, 25)
(53, 8)
(54, 35)
(55, 19)
(56, 11)
(57, 9)
(58, 9)
(59, 22)
(60, 24)
(61, 28)
(62, 21)
(63, 75)
(64, 21)
(65, 31)
(66, 21)
(67, 10)
(68, 8)
(69, 23)
(70, 55)
(71, 24)
(72, 13)
(73, 28)
(74, 24)
(75, 3)
(76, 10)