section15

section15 - 15 Multicommodity tranportation Linear programs...

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15 Multicommodity tranportation Linear programs in practice are often very large. Typically these massive models arise from interfacing smaller models. In this section we describe a typical example. In Section 10 we introduced a model called the “transportation prob- lem”. The model involves transporting a certain commodity from m origins, with supplies s 1 , s 2 , . . . , s m units of the commodity, to n destinations, with demands d 1 , d 2 , . . . , d n units. We assume total supply equals total demand: m i =1 s i = n j =1 d j . To transport one unit of the commodity along the link from origin i to desti- nation j costs c ij , so if we choose to transport x ij units on this link, it costs us c ij x ij . Our problem is to choose nonnegative values for the variables x ij (each of which may also be subject to an upper limit l ij ) in order to minimize the total transportation cost. Mathematically, we can write this problem as follows: minimize m i =1 n j =1 c ij x ij subject to n j =1 x ij = s i ( i = 1 , 2 , . . . , m ) m i =1 x ij = d j ( j = 1 , 2 , . . . , n ) l ij x ij

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