Método Simplex Directo AQUÍ
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Transcript of Método Simplex Directo AQUÍ
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Método Simplex(Directo)
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Método Simplex (directo)
Propiedades:
• Todas las restricciones (con excepción de la no negatividad) deben tener la desigualdad [ y con el término independiente a la derecha y no negativo.
• Todas las variables de decisión deben ser no negativas.
• La función objetivo debe ser Maximizar.
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Algoritmo Simplex
1) Transformar el modelo matemático en formato estándar [F.E]
2) Crear la tabla Simplex y llenarla3) Buscar la solución inicial factible4) Aplicar el método Gauss-Jordan para las
iteraciones5) Verificar si la nueva solución es la óptima de lo
contrario volver al paso 4). 6) Mostrar las solución óptima.
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Tabla Simplex
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Tabla SimplexRenglón
razón
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Tabla SimplexRenglón
razón
-5 -2 0 0 0 0
4 1 1 0 0 200
-2 1 0 1 0 0
4 5 0 0 1 400
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Tabla SimplexRenglón
razón
0 -5 -2 0 0 0 0
1 4 1 1 0 0 200
2 -2 1 0 1 0 0
3 4 5 0 0 1 400
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Tabla SimplexRenglón
razón
0 -5 -2 0 0 0 0
1 4 1 1 0 0 200
2 -2 1 0 1 0 0
3 4 5 0 0 1 400
It =0
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Tabla SimplexRenglón
razón
0 z -5 -2 0 0 0 0
1 4 1 1 0 0 200
2 -2 1 0 1 0 0
3 4 5 0 0 1 400
It =0
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Tabla SimplexRenglón
razón
0 z -5 -2 0 0 0 0
1 4 1 1 0 0 200
2 -2 1 0 1 0 0
3 4 5 0 0 1 400
It =0
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Tabla SimplexRenglón
razón
0 z -5 -2 0 0 0 0
1 4 1 1 0 0 200
2 -2 1 0 1 0 0
3 4 5 0 0 1 400
It =0
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Tabla SimplexRenglón
razón
0 z -5 -2 0 0 0 0
1 4 1 1 0 0 200
2 -2 1 0 1 0 0
3 4 5 0 0 1 400
It =0
CoL umna p i vote
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Tabla SimplexRenglón
razón
0 z -5 -2 0 0 0 0
1 4 1 1 0 0 200
2 -2 1 0 1 0 0
3 4 5 0 0 1 400
It =0
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Tabla SimplexRenglón
razón
0 z -5 -2 0 0 0 0 -
1 4 1 1 0 0 200
2 -2 1 0 1 0 0
3 4 5 0 0 1 400
It =0
𝑅𝑎𝑧 ó𝑛=𝑏𝑖
𝑐𝑜𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑒𝑑𝑒𝑐𝑜𝑙𝑢𝑚𝑛𝑎𝑝𝑖𝑣𝑜𝑡𝑒 [𝑐 .𝑐 .𝑝 .]
𝑅𝑎𝑧 ó𝑛 h1=200
4=5 0
𝑅𝑎𝑧 ó𝑛 h1=0−2
=𝑛𝑜
𝑅𝑎𝑧 ó𝑛 h1=400
4=10 0
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Tabla SimplexRenglón
razón
0 z -5 -2 0 0 0 0 -
1 4 1 1 0 0 200 50
2 -2 1 0 1 0 0 no
3 4 5 0 0 1 400 100
It =0
𝑅𝑎𝑧 ó𝑛=𝑏𝑖
𝑐𝑜𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑒𝑑𝑒𝑐𝑜𝑙𝑢𝑚𝑛𝑎𝑝𝑖𝑣𝑜𝑡𝑒 [𝑐 .𝑐 .𝑝 .]
𝑅𝑎𝑧 ó𝑛 h1=200
4=5 0
𝑅𝑎𝑧 ó𝑛 h1=0−2
=𝑛𝑜
𝑅𝑎𝑧 ó𝑛 h1=400
4=10 0
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Tabla SimplexRenglón
razón
0 z -5 -2 0 0 0 0 -
1 4 1 1 0 0 200 50
2 -2 1 0 1 0 0 no
3 4 5 0 0 1 400 100
It =0
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Tabla SimplexRenglón
razón
0 z -5 -2 0 0 0 0 -
1 4 1 1 0 0 200 50
2 -2 1 0 1 0 0 no
3 4 5 0 0 1 400 100
It =0
Renglón Pivote
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Tabla SimplexRenglón
razón
0 z -5 -2 0 0 0 0 -
1 4 1 1 0 0 200 50
2 -2 1 0 1 0 0 no
3 4 5 0 0 1 400 100
It =0
Renglón Pivote
Elemento Pivote
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Tabla SimplexRenglón
razón
0 z -5 -2 0 0 0 0 -
1 4 1 1 0 0 200 50
2 -2 1 0 1 0 0 no
3 4 5 0 0 1 400 100
It =0
Renglón Pivote
=
𝑅 ′𝑏𝑎𝑠𝑒=[𝑅𝑒𝑛𝑔𝑙ó𝑛𝑝𝑖𝑣𝑜𝑡𝑒] 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑜𝑃𝑖𝑣𝑜𝑡𝑒
𝑅 ′ 0=− (−5 )∗[1 ;0 .25 ;0 .25 ;0 ;0 ;5 0]+[−5 ;−2 ;0 ; 0 ;0 ;0 ] = 𝑅 ′ 2=− (−2 )∗[1 ;0 .25 ; 0 .25 ;0 ;0 ;50 ]+[−2 ;1 ;0 ;1 ;0 ; 0] =
𝑅 ′ 3=− (4 )∗[1 ;0 .25 ; 0.25 ;0 ;0 ;5 0 ]+[4 ;5 ;0 ;0 ;1; 400 ] =
𝑅 ′ 1=[ 4 ;1 ;1;0 ;0 ;2 00]
4
𝑅𝑛𝑢𝑒𝑣𝑜=− (𝑐 .𝑐 .𝑝 )∗𝑅 ′𝑏𝑎𝑠𝑒+𝑅𝑣𝑖𝑒𝑗𝑜
= ;
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Tabla SimplexRenglón
razón
0 z
1
2
3
It =1
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Tabla SimplexRenglón
razón
0 z
1
2
3
It =1
𝑅 ′ 0=[0 ;−0.75 ;1.25 ;0 ;0 ;25 0]
𝑅 ′ 2= [0 ;1.5 ;0.5 ;1 ;0 ;100 ]
𝑅 ′ 3=[0 ; 4 ;−1 ;0 ;1 ;2 00]
𝑅 ′ 1=[1 ;0 .25 ; 0.25 ;0 ;0 ;5 0 ]
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Tabla SimplexRenglón
razón
0 z
1 1 0.25 0.25 0 0 50
2
3
It =1
𝑅 ′ 0=[0 ;−0.75 ;1.25 ;0 ;0 ;25 0]
𝑅 ′ 2= [0 ;1.5 ;0.5 ;1 ;0 ;100 ]
𝑅 ′ 3=[0 ; 4 ;−1 ;0 ;1 ;2 00]
𝑅 ′ 1=[1 ;0 .25 ; 0.25 ;0 ;0 ;5 0 ]
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Tabla SimplexRenglón
razón
0 z 0 -0.75 1.25 0 0 250
1 1 0.25 0.25 0 0 50
2 0 1.5 0.5 1 0 100
3 0 4 -1 0 1 200
It =1
𝑅 ′ 0=[0 ;−0.75 ;1.25 ;0 ;0 ;25 0]
𝑅 ′ 2= [0 ;1.5 ;0.5 ;1 ;0 ;100 ]
𝑅 ′ 3=[0 ; 4 ;−1 ;0 ;1 ;2 00]
𝑅 ′ 1=[1 ;0 .25 ; 0.25 ;0 ;0 ;5 0 ]
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Tabla SimplexRenglón
razón
0 z 0 -0.75 1.25 0 0 250
1 1 0.25 0.25 0 0 50
2 0 1.5 0.5 1 0 100
3 0 4 -1 0 1 200
It =1
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Tabla SimplexRenglón
razón
0 z 0 -0.75 1.25 0 0 250
1 1 0.25 0.25 0 0 50
2 0 1.5 0.5 1 0 100
3 0 4 -1 0 1 200
It =1
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Tabla SimplexRenglón
razón
0 z 0 -0.75 1.25 0 0 250
1 1 0.25 0.25 0 0 50
2 0 1.5 0.5 1 0 100
3 0 4 -1 0 1 200
It =1
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Tabla SimplexRenglón
razón
0 z 0 -0.75 1.25 0 0 250 -
1 1 0.25 0.25 0 0 50 200
2 0 1.5 0.5 1 0 100 66.66..
3 0 4 -1 0 1 200 50
It =1
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Tabla SimplexRenglón
razón
0 z 0 -0.75 1.25 0 0 250 -
1 1 0.25 0.25 0 0 50 200
2 0 1.5 0.5 1 0 100 66.66..
3 0 4 -1 0 1 200 50
It =1
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Tabla SimplexRenglón
razón
0 z 0 -0.75 1.25 0 0 250 -
1 1 0.25 0.25 0 0 50 200
2 0 1.5 0.5 1 0 100 66.66..
3 0 4 -1 0 1 200 50
It =1
=
𝑅 ′𝑏𝑎𝑠𝑒=[𝑅𝑒𝑛𝑔𝑙ó𝑛𝑝𝑖𝑣𝑜𝑡𝑒] 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑜𝑃𝑖𝑣𝑜𝑡𝑒
𝑅 ′ 0=− (−0.75 )∗[0 ;1 ;−0 .25 ;0 ;0 .25 ;50 ]+[0 ;−0.75 ;1.25 ;0 ; 0 ;250] =
𝑅 ′ 1=− (0.25 )∗[0 ;1;−0 .25 ;0 ;0 .25 ;5 0]+[1; 0.25 ;0.25 ;0 ;0 ;5 0] =
𝑅 ′ 2=− (1.5 )∗[0 ;1;−0 .25 ;0 ;0 .25 ;5 0 ]+[0 ;1.5 ;0 .5 ;1 ;0 ;100] =
𝑅 ′ 3=[0 ;4 ;−1 ;0 ;1; 200 ]
4
𝑅𝑛𝑢𝑒𝑣𝑜=− (𝑐 .𝑐 .𝑝 )∗𝑅 ′𝑏𝑎𝑠𝑒+𝑅𝑣𝑖𝑒𝑗𝑜
= ;
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Tabla SimplexRenglón
razón
0 z
1
2
3
It =2
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Tabla SimplexRenglón
razón
0 z
1
2
3
It =2
𝑅 ′ 0 = [0 ;0 ;1.0625 ;0 ; 0.1875 ;287.5]𝑅 ′ 1= [1 ;0 ;0.3125 ; 0 ;−0.0625 ;37.5 ]𝑅 ′ 2= [0 ; 0 ;0.875 ;1 ;−0.375 ;25 ]
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Tabla SimplexRenglón
razón
0 z 0 0 1.0625 0 0.1875 287.5
1 1 0 0.3125 0 -0.0625 37.5
2 0 0 0.875 1 -0.375 25
3 0 1 -0.25 0 0.25 50
It =2
𝑅 ′ 0 = [0 ;0 ;1.0625 ;0 ; 0.1875 ;287.5]𝑅 ′ 1= [1 ;0 ;0.3125 ; 0 ;−0.0625 ;37.5 ]𝑅 ′ 2= [0 ; 0 ;0.875 ;1 ;−0.375 ;25 ]
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Tabla SimplexRenglón
razón
0 z 0 0 1.0625 0 0.1875 287.5
1 1 0 0.3125 0 -0.0625 37.5
2 0 0 0.875 1 -0.375 25
3 0 1 -0.25 0 0.25 50
It =2
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Tabla SimplexRenglón
razón
0 z 0 0 1.0625 0 0.1875 287.5 óptimo
1 1 0 0.3125 0 -0.0625 37.5
2 0 0 0.875 1 -0.375 25
3 0 1 -0.25 0 0.25 50
It =2
𝑆𝑜𝑙𝑢𝑐𝑖 ó𝑛ó𝑝𝑡𝑖𝑚𝑎