transp - OR 320/520 Optimization I Prof Bland Parts of this...

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OR 320/520 Optimization I 10/18/07. Prof. Bland Parts of this handout are adapted from notes of Prof. A.S. Lewis. The Transportation Problem Read: Chapter 3 of the AMPL Book. Recall that at the beginning of the term we discussed examples of the As- signment Problem. (See Lecture Summaries No. 3 and No. 1.) In our first example we sought a minimum cost assignment of 3 chip fabrication facilities to 3 sites. Each fab was to be assigned to exactly one site and each site was to have exactly one fab assigned to it. The cost of assigning fab i site j was given as c ij for each choice of i = 1 , 2 , 3 and j=1 , 2 , 3. We defined decision variables x ij and wrote an integer linear programming model: minimize 3 X i =1 3 X j =1 c ij x ij subject to 3 X j =1 x ij = 1 ( i = 1 , 2 , 3) 3 X i =1 x ij = 1 ( j = 1 , 2 , , 3) x ij { 0 , 1 } ( i, j = 1 , 2 , 3) . For each i, j ∈ { 1 , 2 , 3 } we interpret x ij = 1 to mean fab i is assigned to site j and we interpret x ij = 0 to mean fab i is not assigned to site j . The first set of constraints forces each fab to be assigned to one site and the second set forces each site to have exactly one fab assigned to it. The objective function gives the total cost of a feasible assignment. The input for our fabs and sites example was a 3 × 3 cost matrix c and the output can be regarded to be an assignment of each row to exactly one column and each column to exactly one row at minimum total cost. In its general form the input to the Assignment Problem is an n × n cost matrix c and we want to assign each row to exactly one column and each column to exactly one row
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