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Unformatted text preview: Homework 6 Solution March 10, 2009 Solution to Exercise 1, page 197 (a) False. To balance the transportation model, we need either a dummy source or a dummy destination; but not both. (b) True. (c) True. Solution to Exercise 5, page 198- Just set up the model. We have three sup- ply centers LA, Detroit and New Orleans and a “dummy supply node” due to the demand exceeding the total capacity of all supply nodes. We can label these nodes respectively, from 1 through 4. Their supply level is a 1 = 1000 , a 2 = 1300 , a 3 = 1200 , a 4 = 200 . Note that the supply at the dummy nodes is equal to the difference of the total demand (3700) and the total supply (3500). There are two demand nodes, Denver and Miami, and we label them by 1 and 2, respectively. Their demand values are b 1 = 2300 , b 2 = 1400 . We introduce the flow variables: (1) x 11 representing the flow from LA to Denver (since LA does not deliver to Miami, there is no other variable for this supply node.) (2) x 21 and x 22 representing the flow from Detroit to Denver and Miami, respec- tively. (3) x 31 and x 32 representing the flow from New Orleans to Denver and Miami, respectively. (4) x 41 and x 42 representing the flow from the “dummy supply node” to Denver and Miami, respectively. 1 Using the transportation costs given in Table 5.2 on page 195, and the fact that penalties (costs actually) for undelievered cars to Denver and Miami and $200 and $300, respectively, we have the following objective: minimize 80 x 11 + 100 x 21 + 108 x 22 + 102 x 31 + 68 x 32 + 200 x 41 + 300 x 42 . The balance equations for supply nodes are x 11 = 1000 ( LA ) x 21 + x 22 = 1300 ( Detroit ) x 31 + x 32 = 1200 ( New Orleans ) x 41 + x 42 = 200 ( Dummy ) ....
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This note was uploaded on 08/31/2010 for the course IESE GE 330 taught by Professor Nedich during the Spring '09 term at University of Illinois at Urbana–Champaign.
- Spring '09