# In the node–arc formulation of the ‘‘traffic-assignment’’ problem given in Exercise 15, the subproblems correspond to finding the shortest path between the kth origin–destination

In the node–arc formulation of the ‘‘traffic-assignment’’ problem given in Exercise 15, the subproblems correspond to finding the shortest path between the kth origin–destination pair. In general, there may be a large number of origin– destination pairs and hence a large number of such subproblems. However, in Chapter 11 on dynamic programming, we saw that we can solve simultaneously for the shortest paths from a particular origin to all destinations. We can then consolidate the subproblems by defining one subproblem for each node where traffic originates. The conservation of-flow constraints become: where the summation is the total flow emanating from origin s for all destination nodes. In t denotes the total flow on arc i − j that emanates from origin s; that is, the summation is carried over all origin–destination pairs k = (s, t) whose origin is node s. a) How does the decomposition formulation developed in Exercise 15 change with this change in definition of a subproblem? Specify the new formulation precisely.

b) Which formulation has more constraints in its restricted master?

c) Which restricted master is more restricted? [Hint. Which set of constraints implies the other?]

d) How does the choice of which subproblems to employ affect the decomposition algorithm? which choice would you expect to be more efficient? Why?