GE330_lect11

GE330_lect11 - Lecture 11 Transportation Model and Its...

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Lecture 11 Transportation Model and Its Variants March 4, 2009

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Deﬁnition of the Transportation Model Given m sources and n destinations, the supply at source i is a i and the demand at destination j is b j . The cost of shipping one unit of goods from source i to destination j is c ij . The goal is to minimize the total transportation cost while satisfying all the supply and demand restrictions. 2
General Model Decision variables: x ij the amount shipped from source i to destination j . Objective function: minimize m i =1 n j =1 c ij x ij . Constraints – supply: for each source node i , n j =1 x ij = a i ; – demand: for each destination node j , m i =1 x ij = b j . The general transportation model: min m X i =1 n X j =1 c ij x ij s.t. n X j =1 x ij = a i , i = 1 , 2 , ··· ,m m X i =1 x ij = b j , j = 1 , 2 , ··· ,n x ij 0 , i = 1 , 2 , ··· ,m ; j = 1 , 2 , ··· ,n (1) 3

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An Example MG Auto has three plants in Los Angeles, Detroit, and New Orleans, and two major distribution centers in Denver and Miami. The capacity of the three plants during the next quarter are 1000, 1500, and 1200 cars. The quarterly demand in the two distribution centers are 2300 and 1400 cars. The transportation costs per car on the diﬀerent routes is given in the following table. Denver(1) Miami(2) Los Angeles(1) \$80 \$215 Detroit(2) \$100 \$108 New Orleans(3) \$102 \$68 NOTE: In practice, some of the links might not be available. To deal with this situation, we can assign a very big cost on the unavailable links so that the ﬂow on them are zero in the optimal solution. 4
The LP model min z = 80 x 11 + 215 x 12 + 100 x 21 + 108 x 22 + 102 x 31 + 68 x 32 s.t. x 11 + x 12 = 1000 x 21 + x 22 = 1500 x 31 + x 32 = 1200 x 11 + x 21 + x 31 = 2300 x 12 + x 22 + x 32 = 1400 x ij 0 , i = 1 , 2 , 3 ,j = 1 , 2 The optimal solution is shown below 5

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Balancing the Model In the transportation model, the LP is feasible only if the model is balanced, i.e., the total demand equals to the total
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GE330_lect11 - Lecture 11 Transportation Model and Its...

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