laboratorio_viernes_8_abril - Laboratorio 5 Optimizacin...

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Laboratorio 5 Optimizaci´on: Programaci´on Lineal Viernes 8 de Abril 2011 1. etodo Simplex Proposici´ on 1.1 . Sea x una soluci´on factible b´asica de ( PL ) asociada a la base B . Supongamos que ( g PL B ) es la forma expl´ ıcita de ( PL ) respecto a B . Se cumple que Si para todo j I N los coeficientes ( z j - c j ) 0, entonces x es una soluci´on optimal de ( PL ). Si existe j 0 I N tal que ( z j - c j ) > 0 y para todo i I B los coeficientes Y ij 0 0, entonces d T = ( - Y T · j 0 , e T j 0 ) es una direcci´on infinita de ( PL ) y la funci´on objetivo z no es acotada inferiormente sobre el conjunto factible de ( PL ). Proposici´ on 1.2 . Sea x una soluci´on factible b´asica de ( PL ) asociada a la base B y ( g PL B ) la forma expl´ ıcita correspondiente. Supongamos que ( z j 0 - c j 0 ) > 0, para cierto j 0 I N fijo y que Y · j 0 6≤ 0. Entonces La funci´on objetivo puede mejorarse aumentando la variable no b´asica x j 0 desde cero hasta el valor ¯ x j 0 = ¯ x i 0 Y i 0 j 0 = min ¯ x B i Y ij 0 ; i I B , Y ij 0 > 0 (1) Si intercambiamos en B la columna B · i 0 por A · j 0 , se obtiene una nueva base B 0 . La soluci´on b´asica x 0 asociada a B 0 es factible y tal que x 0 i 0 = 0, x 0 j 0 = ¯ x j 0 y c T x 0 c T x . 1.1. Ejemplo Resolver mediante el algoritmo Simplex. min z = - 2 x 1 + 5 x 2 - x 3 s. a: 2 x 1 + 3 x 2 - x 3 3 - x 1 + 2 x 2 + x 3 4 x 1 + 3 x 2 9 x 1 , x 2 , x 3 0 1
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Soluci´ on 1. Escribir en forma est´andar.
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