ESI 6912 Fall 2008
Advanced Network Optimization Ravindra K. Ahuja
Assignment # 8
Problem 1
Consider an extension of the multicommodity flow problem with nonhomogeneous goods. In
this model, each unit
k
ij
x
of flow of commodity k on arc (i, j) consumes a given amount
k
ij
of the
capacity (or some other resource) associated with the arc (i, j), and we replace the bundle
constraint with a more general resource availability constraint
1
k k
ij ij ij
k K
x u
.
(a) Show how to convert this model into the bundle constraint model (17.1) that we have
considered in this chapter if each commodity consumes the same amount of resource on each arc,
that is, for every arc (i, j),
1 2
...
K
ij ij ij ij
.
(b) Show how to solve the general version of this problem by using a modification of the
following methods that we have considered in this chapter: (i) Lagrangian relaxation, and (ii)
column generation.
Problem 2
Suppose that you had an algorithm that could find a feasible multicommodity flow for the
problem (17.1). Further, suppose that you also had the optimal set of prices (tolls) for bundle
constraints for the arcs. How might you use these prices and the solution algorithm to find an
optimal flow for the multicommodity flow problem?
Problem 3
Show that the multicommodity flow problem always has K redundant constraints by viewing it
as follows. Each commodity flows on a separate, but identical network and the bundle
constraints tie together the flows on the separate network. In this "layered representation" of the
problem, let jk denote the copy of node j on the network (layer) corresponding to commodity k.
Next show that it is possible to replace the nodes 1k for k = 1 to K by a single node 1*, and that
the resulting formulation has a connected network.
Problem 4
Show how to apply column generation and Dantzig-Wolfe decomposition to multicommodity
flow problems in which each commodity can have several sources and destinations. Assume that
the cost vector for each commodity is nonnegative. For each commodity k, let xk,q for q = 1,
2, ..., Qk denote the flow vector corresponding to the qth of Qk feasible spanning tree solution for
the system Nxk= bk, 0 ≤ xk ≤ uk. Use the fact that we can write any potentially optimal feasible
solution xk to this system as xk = 1≤k≤K k,q xk,q for some nonnegative weighting vectors k,q
1
