Wireless Network Design: Optimization Models and Solution ...

8 The Design of Partially Survivable Networks 187
the R1 and H1 nodes, respectively; constraints 8.20 and 8.21 are flow balance constraints
on the intermediate R j and Hj (t = 2, ...,(T − 1)) nodes, respectively;
constraints 8.22 and 8.23 are flow balance constraints on the RT and HT nodes,
respectively; constraint 8.25 indicates flow into Q.
One way in which the node weights could be incorporated into this formulation
is by adding constraints y1 ≥ zPH 1 for time period 1 and y j ≥ zR ( j−1) Hj +zH ( j−1) H j for
the other time periods, and adding the term (+ T
∑ e jy j) into the objective function
j=1
(e j is the weight at node Hj). However, since at most one of the edges coming into
an Hj node will be set to 1, we can add the node weights to the cost of every edge
coming into Hj and solve (S1) with these modified edge weights to get the optimal
solution to the problem. This implies that we can solve a shortest path problem to
solve the subproblem. The reduced cost can then be obtained by subtracting the dual
price πi associated with cell i from the subproblem objective function value.
8.3.2.3 Integer Programming and Column Generation
The most common technique for solving integer programs is branch and bound
(Land and Doig [10]). In a column generation framework, switching on the integrality
requirements after solving the LP relaxation usually does not guarantee the
optimal integral solution. However in many cases, if enough good quality columns
have been generated, switching on the integrality requirements and solving the resulting
integer program gets us very close to the integer optimum. We have used this
approach in this paper, as the gaps are observed to be very small.
Exact column generation procedures for integer programs are relatively more
difficult. The difficulty usually lies in the development of an effective way to incorporate
the linear programming column generation into a branch and bound scheme.
However, a considerable volume of work has focused on this general branch-andprice
approach (e.g., Barnhart et al. [2], Vance et al. [18]). Here, we briefly outline
two valid branching schemes for branch and price.
Branching Scheme I
Identify a fractional variable vs1; let i be the cell associated with this variable. From
constraint set 8.14, we can infer that there is at least one other variable involving
this cell that is fractional. Find one such variable, say vg2. As there are no identical
columns, there is at least one period where these columns differ. Find one such
period, say t. A valid branching scheme is one which branches on cell i being connected
to the hub in period j, or not. Under this scheme, the subproblems at the
branches would be solved as follows. If on the branch where i is connected to the
hub in j, the edges of the subproblem which allow for connecting this cell to the
MTSO (i.e., zR ( j−1) R j and zH ( j−1) R j ) are set to 0. For all cells k �= i, no alteration is
needed in the subproblem. If on the branch where i is connected to the MTSO in