L20_Transportation problems - Formulations

# L20_Transportation problems - Formulations - The...

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Unformatted text preview: The Transportation Model – Formulations The Transportation Model The transportation model is a special class of LPPs that deals with transporting(=shipping) a commodity from sources (e.g. factories) to destinations (e.g. warehouses). The objective is to determine the shipping schedule that minimizes the total shipping cost while satisfying supply and demand limits. We assume that the shipping cost is proportional to the number of units shipped on a given route. We assume that there are m sources 1,2, …, m and n destinations 1, 2, …, n. The cost of shipping one unit from Source i to Destination j is c ij . We assume that the availability at source i is a i (i=1, 2, …, m) and the demand at the destination j is b j (j=1, 2, …, n). We make an important assumption: the problem is a balanced one. That is ∑ ∑ = = = n j j m i i b a 1 1 That is, total availability equals total demand. We can always meet this condition by introducing a dummy source (if the total demand is more than the total supply) or a dummy destination (if the total supply is more than the total demand). Let x ij be the amount of commodity to be shipped from the source i to the destination j. Thus the problem becomes the LPP ∑ ∑ = = = n j ij ij m i x c z 1 1 Minimize subject to ) ,..., 2 , 1 ( ) ,..., 2 , 1 ( 1 1 n j b x m i a x j m i ij i n j ij = = = = ∑ ∑ = = ≥ ij x Thus there are m × n decision variables xij and m+n constraints. Since the sum of the first m constraints equals the sum of the last n constraints (because the problem is a balanced one), one of the constraints is redundant and we can show that the other m+n-1 constraints are LI. Thus any BFS will have only m+n-1 nonzero variables. Though we can solve the above LPP by Simplex method, we solve it by a special algorithm called the transportation algorithm. We present the data in an m × n tableau as explained below. c 11 c 12 c 1n a 1 c 21 c 22 c 2n a 2 c m1 c m2 c mn a m b 1 b 2 b n S o u r c e 1 2 . . m Destination 1 2 . . n Supply Demand Formulation of Transportation Models Example 5.1-2 MG Auto has three plants in Los Angeles, Detroit, and New Orleans, and two major distribution centers in Denver and Miami. The capacities of the three plants during the next quarter are 1000, 1300...
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L20_Transportation problems - Formulations - The...

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