HW #2
ME 391Q
Algorithms for MIPs
ODIMCF Problem Formulation
The origin-destination integer multicommodity flow problem (ODIMCF) is defined over
the network G = (N, A) comprised of node set N and arc set A. In this version of the problem, the
flow of a commodity (defined in this case by the origin-destination pair) may use only one path
from origin to destination. Let K be the set of commodities that are to be routed on the network.
ODIMCF contains binary decision variables x =  xijk  , where xijk = 1 if the entire quantity
(denoted by qk) of commodity k is assigned to arc ij and 0 otherwise. The cost of assigning
commodity k in its entirety to arc ij is equal to qk  cijk , where cijk is the unit flow cost for arc ij.
Arc ij has capacity dij, for all ij  A; node i has a supply of commodity k (denoted by bik ) equal to
1 if i is the origin node for k, equal to –1 if i is the destination node for k, and equal to 0
otherwise. (There is only one source node and one destination node for each commodity.)
The conventional node-arc ODIMCF formulation is as follows.
Minimize
 c q
kK ijA
subject to
k
ij
q x
k
kK
k
ij
k
xijk
 dij ,  ij  A
x x
ijA
k
ij
(1)
jiA
k
ji
(2)
 bik ,  k  K , i  N
xijk 0,1,  ij  A,  k  K
(3)
(4)
Note that without restricting generality of the problem, the arc flow variables xijk are modeled as
binary variables. To do this, it was necessary to scale the supply and demand (bi = 1) for each
commodity to 1 and accordingly adjust the coefficients in the objective function (1) and in
constraints (2).
Assignment:
a. Construct a small example with 2 commodities and 5 nodes to illustrate the problem. Include
data, draw the network flow diagram, and indicate the solution.
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b. Develop a path-based formulation that can be used in a column generation algorithm.
In the
formulation, there should again be an underlying network G comprised of node set N and arc
set A, with qk representing the quantity of commodity k. Let P(k) represent the set of all
origin-destination paths in G for commodity k. In your model, denote the binary decision
variables by y kp , where y kp = 1 if all qk units of commodity k are assigned to path p  P(k)
and 0 otherwise. Let the cost of assigning commodity k in its entirety to path p equal qk
times the unit flow cost for path p. Denote this cost by c kp (what is this value in terms of the
original data?) As before, arc ij has capacity dij, for all ij  A. Finally, let the parameter  ijp
= 1 if arc ij is contained in path p  P(k) for some k  K and 0 otherwise. Discuss the
purpose of each constraint in the model.
c. Formulate the pricing subproblem that must be solved to generate columns. Introduce
additional notation as necessary. What is the generic name of the subproblem? What does
the solution to the subproblem tell you about the strength of the bound provided by the LP
relaxation of the path-based model?
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