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Unformatted text preview: 15 Multicommodity transportation Linear programming problems in practice are often very large. Typically these massive models arise from interfacing smaller models. In this section we describe a typical example. For more, look at Chapter 4 in the AMPL book . In Section 11 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 X i =1 s i = n X j =1 d j . To transport one unit of the commodity along the link from origin i to destination 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 X i =1 n X j =1 c ij x ij subject to n X j =1 x ij = s i ( i = 1 , 2 , . . . , m ) m X i =1 x ij = d j ( j = 1 , 2 , . . . , n ) ≤ x ij ≤ l ij ( i = 1 , 2 , . . . , m, j = 1 , 2 , . . . , n ) ....
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- Linear Programming, cij xij, AMPL book, GARY CLEV PITT, cijp xijp