These results are used to devise an algorithm that generates all non-dominated higher-order cover cuts, and, in particular, to formulate and solve suitable separation problems for deriving a higher-order cut that deletes a given fractional solution to an underlying continuous relaxation.
For a given fractional solution (x ^, z ^ ), we set the capacities to x ^ ij, for all ij ∈ A. We then calculate the maximum flow that can be sent from r to each of the customers j ∈ R. If there exists customer j such that the value of the maximum flow is less than one, we obtain a cut set, say W ⊂ V, r ∈ W, such that capacity of δ + (W ) is less than one.
The traditional separation procedure solves a Max-Flow-Min-Cut problem on a graph reflecting the current fractional solution.
Let (u ¯, x ¯, z ¯ ) be the current fractional solution at a given node of the search tree and, for all d ∈ {3,4,…, m}, consider a pair of vertices i, j∈ Q d.
These constraints are also used as cutting planes, that is, we already add them in a cutting plane callback when they are violated by the fractional solution.
Figure 1 Entropy fractional solution.
The reduced linear programs are then solved by the ILOG CPLEX LP solver, and in cases where fractional solutions are found, an ILP solver is applied.
The remaining instances with fractional solutions were easily solved by the ILP solver.
Assume that (Phi _{alpha } (t^{alpha } )) is a fractional fundamental matrix solution of (26).
