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# sol7 - ,3,4(5000,5(1000,0 u 1 u 2 v 1 v 2 v 3 v 4 =(0 3 5 4...

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AMS 341 Spring, 2010 Homework Set # 7 — Solution Notes #1. Let ( - ,y ) =(-,cost). The balanced formulation: Vendor Salary Personnel Dummy Site 1 (-,5) (-,4) (-,2) (-,0) 10000 Site 2 (-,3) (-,4) (-,5) (-,0) 6000 5000 5000 5000 1000 (a) Northwest cornor method: 5000 5000 6000 X 5000 5000 1000 = 5000 5000 0 6000 X X 5000 1000 = 5000 5000 0 X 6000 X X 5000 1000 = 5000 5000 0 X 5000 1000 X X X 1000 = 5000 5000 0 X 5000 1000 X X X X X (b) Min-cost method: 1000 9000 6000 5000 5000 5000 X = 5000 1000 4000 6000 5000 5000 X X = 5000 1000 4000 5000 1000 X 5000 X X = 4000 5000 1000 X 5000 1000 X 1000 X X = 4000 5000 1000 X 5000 1000 X X X X X (c) Vogel’s method: 5000 5000 2 6000 3 5000 5000 X 1000 2 0 3 0 = 5000 1000 4000 4 6000 3 5000 5000 X X 2 0 - 0 = 5000 1000 4000 1 5000 1000 1 X 5000 X X 2 0 - - = 4000 5000 1000 X 4 5000 1000 4 X 1000 X X - 0 - - = 4000 5000 1000 X - 5000 1000 X 4 X X X X - 4 - - #2. Transportation simplex method: (-,5) (4000,4) (5000,2) (1000,0) (5000,3) (1000,4) (-,5) (-,0) ( u 1 ,u 2 ,v 1 ,v 2 ,v 3 ,v 4 , c 11 , c 23 , c 24 ) = (0 , 0 , 3 , 4 , 2 , 0 , - 2 , - 3 , 0), it is optimal. #3. Transportation simplex method: (5000,5) (5000,4) (0,2)
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Unformatted text preview: (-,3) (-,4) (5000,5) (1000,0) ( u 1 , u 2 , v 1 , v 2 , v 3 , v 4 ) = (0 , 3 , 5 , 4 , 2 ,-3), we check u 2 + v 1 > c 21 , so x 21 enters the basis, x 11 and x 23 decrease, x 13 increases by 5000 and we get the next table: (-,5) (5000,4) (5000,2) (-,0) (5000,3) (-,4) (0,5) (1000,0) ( u 1 , u 2 , v 1 , v 2 , v 3 , v 4 ) = (0 , 3 , , 4 , 2 ,-3), we check u 2 + v 2 > c 22 , so x 22 enters the basis, x 12 and x 23 decrease, x 13 increases by 0 and we get the next table: (-,5) (5000,4) (5000,2) (-,0) (5000,3) (0,4) (-,5) (1000,0) ( u 1 , u 2 , v 1 , v 2 , v 3 , v 4 ) = (0 , , 3 , 4 , 2 , 0), we check and all non basic variables satisfy u i + v j ≤ c ij so we are done....
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