Each iteration of inner decomposition of the AD CSP problem involves finding the single best feasible path for a fixed defense plan in the inner sub problem. That path is attacked in the inner decomposition AD master problem. Inner decomposition could be done more quickly if multiple feasible short paths were fed to the master problem instead of just one path at a time. When the Lagrangian relaxation of the inner AD sub problem is iterated and the Lagrangian bounds do not match, there is an opportunity to find multiple feasible paths to close the Lagrangian optimality gap. The previous section described how path enumeration is used to close the gap between the Lagrangian upper and lower bounds of the relaxation of the AD sub problem by finding the shortest feasible path that exists between the Lagrangian upper and lower bounds. But, path enumeration also generates all of the feasible paths that have a cost between the Lagrangian upper and lower bounds of the relaxed AD inner sub problem. Instead of only utilizing the shortest cost path in between the Lagrangian upper and lower bounds found in the inner sub problem, we can utilize any subset of the feasible paths generated by the path enumeration algorithm in an attempt to reduce the number of iterations in the inner AD sub problem. Each feasible path generated through the path enumeration algorithm becomes an input cut to the inner decomposition AD master problem. Our use of more than one path per iteration as input cuts to the AD master problem are called multi-cuts.
In order to implement multi-cuts into the path enumeration algorithm, we only need to make one change in the final step of the path enumeration algorithm:
Return all time budget feasible paths with their associated costs.
This change means the CSP with Lagrangian relaxation inner AD master problem receives many feasible paths as inputs during an iteration. All feasible paths can now be given as input cuts to the AD master problem.
However, the use of multi-cuts does have a shortcoming. In large networks, it may be possible that an enormous number of feasible paths exist with costs that are between the Lagrangian upper and lower bounds. If the upper and lower bound gap is significantly large, an enormous amount of paths could become input cuts. We do not recommend the use of all multi-cuts on large networks since the constraint set could become prohibitively large. Instead, an analyst could limit the size of the constraint set by selecting only small number of multi-cuts to pass to the master problem.
Another criticism of the use of multi-cuts as inputs to the inner AD master problem is that using all feasible paths at once could generate some superfluous cuts in the AD master problem that do not influence the choice of the optimal attack solution. A rebuttal to this criticism is that it is also possible that some relevant cuts that would have been found in later iterations of inner decomposition may be discovered in earlier inner iterations because of the path enumeration routine. Therefore, we believe the benefit of finding relevant cuts earlier than normal through path enumeration outweighs the possibility of including some useless path cuts in the inner AD master problem.
Consider the six node test network. In this example, we assume zero defense budget, so no arcs are defended. We assume an attack budget of one unit, so only one arc may be attacked. We assume a time budget of 14 units. Next, we perform decomposition to obtain the best attack and the resulting shortest cost path subject to the time budget.
Lagrangian relaxation with path enumeration on the six node test network will take three iterations to find the worst-case attack and shortest path cost. The details of each iteration of Lagrangian relaxation with path enumeration are shown in Table 30. Figure 48 shows the final results of each iteration implemented on the test network. In Figure 48, each iteration is shown, starting in the top left. The final attack is shown with a red “X” and the shortest path chosen is a blue line for each iteration. The lower bound is obtained by the cost of the path with the initial attack, and the upper bound is obtained by the cost of the path with the final attack.
Table 30. Lagrangian Relaxation and Path Enumeration on Test Network Iteration Initial
Attack Shortest Path
Final Attack AD Upper Bound AD Lower Bound 1 None (1,3), (3,2), (2,4), (4,6) (1,3) 38 13 2 (1,3) (1,2), (2,4), (4,5), (5,6) (2,4) 38 14 3 (2,4) (1,3), (3,2), (2,5), (5,6) (2,4) 15 15
First Iteration Second Iteration
Third Iteration
Figure 48. Lagrangian Relaxation and Path Enumeration on Test Network
Next, we examine the effect of introducing multi-cuts on the six node test network. The details of each iteration of Lagrangian relaxation with multi-cut path enumeration are shown in Table 31. Figure 49 shows the final results of each iteration implemented on the test network. In Figure 49, the first and second iterations are shown
from left to right. In the first iteration, the three feasible paths obtained from enumeration are shown with a solid blue line, a short dashed purple line, and a longer dashed orange line. The final attack based on all multi-cuts is shown with a red “X.” The second iteration only has one feasible path remaining.
Table 31. Multi-cut Algorithm Improvement on Six Node Test Network Iteration Initial
Attack Feasible Paths
Final Attack AD Upper Bound AD Lower Bound 1 None (1,3), (3,2), (2,4), (4,6) (1,2), (2,4), (4,5), (5,6) (1,3), (3,2), (2,5), (5,6) (2,4) 15 13 2 (2,4) (1,3), (3,2), (2,5), (5,6) (2,4) 15 15
(a) First Iteration (b) Second Iteration
Figure 49. Multi-cut Algorithm Improvement on Six Node Test Network
The beneficial result of the inclusion of multi-cuts on the example problem instance is the number of iterations required to find the solution to the inner AD decomposition is reduced. The first iteration is able to enumerate three feasible paths before an attack is determined by the master problem. The worst-case attack can only affect two of the three feasible paths. When comparing the first iteration of the multi-cuts improvement with the three iterations of inner decomposition without the multi-cuts improvement, it can be seen that all feasible paths are obtained in one step with multi- cuts, and in this case, no superfluous paths were created. The multi-cuts improvement
eliminated the need to perform Lagrangian relaxation three times in order to obtain the three relevant paths for the AD master problem. There is only one feasible path found by enumeration in the second iteration because the other feasible paths from the first iteration are excluded. The penalty (q) imposed by the initial attack raises the cost of arc (2, 4) by an additional 25 units. The penalized cost of the arc results in paths that are more expensive than the upper bound.