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help, someone can help me? Minimum cost flow with fixed costs and awards for strings saturated.

Consider the following variant of the problem of minimum cost flow where in addition to the network G = (V, A) with values bi associated with nodes i ∈ V, such that Pi∈V bi = 0 and costs cij for the unit cost of transport along the arc (i, j) ∈ A we also have that:

• in each arch is associated with a capacity value that indicates the maximum flow dij transportable along the arc; • the number of arcs along here has sent a strictly positive flow is no more than a percentage 100p1% of the total of the arches and for each of these arcs you pay a fixed cost of K; • the number of arcs that are saturated (arcs along which is sent a flow equal to their capacity) is at least a percentage 100p2% of the total of the arches (p2

Formulate the mathematical model for this problem, it is written in AMPL and defining the data of a particular instance, resolving it. Care must also be an analysis of what happens if you change some of the instance data. In particular, you may find an interval [p1, p2] as small as possible so that there is a solution of the problem.

gab
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    This seem to be a copy paste of a homework assignment. Have you actually tried to solve this at all using AMPL? If you have, please edit and add that code into your question. What is bi? Is Pi an element of the vertice set V? – Cenderze Feb 06 '15 at 07:16

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I'm not sure that I clearly understand your problem, I try to give a possible solution for each question:

  • You should have a positive variable let's call it Xij for each arc, which defines the current flow passing on the arc between nodes i and j.

  • With this variable and the given parameter Dij you can add a constraint to express the capacity boundary: Xij<=Dij ForEach (i,j) belonging to A.

  • About the others constraint I suggest you to use a minimization objective function of the sum{ i in N , j in N } used[i,j] * k . Where used[i,j] is a binary variable that denotes if the corresponding flow is equal to zero or not.To relate the flow with this binary variable you should add an additional constraint like this:

x[i,j] <= d[i,j] * used[i,j]

  • As far the number of saturated arcs concern, you can solve the max flow problem, in which the solution is given by consecutive iterations of the augmenting flow algorithm.

I'm not sure that I answer to your questions if I don't feel free of posting exactly what is your decision problem ( which is the objective function and what are the constraints )

LNRD.CLL
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