Assignment4_solution

# Assignment4_solution - MIT 2.098/6.255/15.093 Optimization...

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MIT 2.098/6.255/15.093 Optimization Methods Prof. J. Vera, Fall 2007 Homework Assignment 4. Solution Guidelines The purpose of this assignment is to give you the opportunity to solve a “real” problem using available software, and to improve the solution procedure. The problem is coded in AMPL and you may use the student edition of AMPL for this. The pointers to obtain it are in one of the announcements of the class. I do want, also, that you try some optimization resources available in the web. You will need to do some extensions to the AMPL code, but they are relatively straight- forward, you can mail us with speciﬁc questions on AMPL. Problem description: The problem is the already known network design problem, but in this case it applies to a multi- commodity ﬂow problem. So, Assume we have a network ( N,A ), where the arcs will have to be actually built to satisfy some give demand between a certain number of origin - destination pairs. There are K of this pairs of nodes. Let o k = ( r k ,s k ) denote the origin-destination pair k , r k N,s k N . For each pair k there is a demand of ﬂow, d k which needs to travel from r k to s k . Each arc has a capacity u ij . For each arc there is a unit cost c ij for traveling arc ( i,j ) (we make it independent of the origin-destination pair in this problem). There is a construction cost G ij if we decide to use arc ( i,j ) A . We model this with a variable x k ij , corresponding to the ﬂow in arc ( i,j ) corresponding to the origin-destination k , and with a binary variables z ij which equals 1 if we construct the arc and 0 otherwise. The model is: min k K ( i,j ) A c k ij x k ij + ( i,j ) A G ij z ij s.t. j :( i,j ) A x k ij - p :( p,i ) A x k pi = b k i ,i N,k = 1 ,...,K K k =1 x k ij u ij z ij , ( i,j ) A x ij 0 ,z ij ∈ { 0 , 1 } , ( i,j ) A where b k i = d k , i = r k - d k , i = s k 0 otherwise To work with this problem, you have to download the ﬁles “net1v1mult.mod” and “net1v1mult.dat”. They contain the model and the basic data. To introduce variation into the diﬀerent data instances you will solve, you will all use diﬀerent sets of data in which basically what changes is the cost infor- mation. To obtain the data which you should use, you have to download the Excel ﬁle “DATA.XLS”, generate the data, and copy it in the corresponding places in the ﬁle “net1v1mult.dat”, follow the instructions in the Excel ﬁle.

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## Assignment4_solution - MIT 2.098/6.255/15.093 Optimization...

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